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Bounds on the entries of the principal eigenvector of the distance signless Laplacian matrix
Linear Algebra and its Applications 483 (2015) 200–220
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Bounds on the entries of the principal eigenvector of the distance signless Laplacian matrix Kinkar Ch. Das a,∗,1 , Celso M. da Silva Junior b,2 , Maria Aguieiras A. de Freitas c,2 , Renata R. Del-Vecchio d,2 a
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea b PEP-COPPE, Universidade Federal do Rio de Janeiro and Centro Federal de Educação Tecnológica, Rio de Janeiro, Brazil c Instituto de Matemática and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil d Instituto de Matemática, Universidade Federal Fluminense, Rio de Janeiro, Brazil
a r t i c l e
i n f o
Article history: Received 31 October 2014 Accepted 2 June 2015 Available online xxxx Submitted by D. Stevanovic MSC: 05C12 05C50 15A18 Keywords: Distance signless Laplacian matrix Spectral radius Principal eigenvector Independence number Diameter
a b s t r a c t The distance signless Laplacian spectral radius of a connected graph G is the largest eigenvalue of the distance signless Laplacian matrix of G, defined as DQ (G) = T r(G) + D(G), where D(G) is the distance matrix of G and T r(G) is the diagonal matrix of vertex transmissions of G. In this paper we determine upper and lower bounds on the minimal and maximal entries of the principal eigenvector of DQ (G) and characterize the extremal graphs. In addition, we obtain a lower bound on the distance signless Laplacian spectral radius of G based on its order and independence number, and characterize the extremal graph. © 2015 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: [email protected] (K.Ch. Das), [email protected] (C.M. da Silva Junior), [email protected] (M.A.A. de Freitas), [email protected]ff.br (R.R. Del-Vecchio). 1 Work supported by National Research Foundation of Korea with the grant no. 2013R1A1A2009341. 2 Work partially supported by CNPq and FAPERJ-Brasil. http://dx.doi.org/10.1016/j.laa.2015.06.003 0024-3795/© 2015 Elsevier Inc. All rights reserved.
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1. Introduction All graphs considered in this paper are finite, simple and connected. Let G = (V, E) be a graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G), where |V (G)| = n and |E(G)| = m. Let deg G (vi ) denote the degree of vertex vi in G. The maximum vertex degree is denoted by Δ and the minimum by δ. The graph G is called degree regular, or simply regular, if all vertices of G have the same degree. Let the distance (the length of a shortest path) between vertices vi and vj of G denoted by di, j . Denote by d the diameter of graph G. Given a graph G, a subset S of V (G) is said to be an independent set of G if the subgraph G[S], induced by S, is a graph with |S| isolated vertices. The independence number α = α(G) of G is the number of vertices in the largest independent set of G. The transmission T r(vi ) or simply Ti , of a vertex vi is defined as the sum of the distances from vi to all other vertices in G, i.e., T r(vi ) = di, j . We denote by T and t the vj ∈V (G)
maximal and minimal transmissions of G, respectively, that is, T = max 1≤i≤n T r(vi ) and t = min 1≤i≤n T r(vi ). The graph G is called k-transmission regular if T r(vi ) = k for every vertex vi ∈ V (G). The distance matrix of G, denoted by D(G), is the n × n matrix whose (i, j)-th entry is equal to di, j , i, j = 1, 2, . . . , n. For more details about the distance matrix we recommend [1]. In [2], Aouchiche and Hansen introduced the distance signless Laplacian matrix of a connected graph G as DQ (G) = T r(G) + D(G), where T r(G) is the diagonal matrix of vertex transmissions of G. This matrix has been studied by several researchers and recent results can be found in [7,8,10,12,13]. For a connected graph G the adjacency, signless Laplacian, distance and distance signless Laplacian matrices are symmetric, nonnegative and irreducible. Therefore, by the Perron–Frobenius theorem [9], fixed p (1 ≤ p < ∞), each of these matrices has a unique eigenvector Y = (y1 , y2 , . . . , yn )T , positive and unitary (Yp = 1) associated with its spectral radius. This eigenvector is called the p-normalized principal eigenvector of the matrix. The study of the principal eigenvector has aroused interest, as evidenced by the articles mentioned below. In 2000, Papendieck and Recht [11] obtained an upper bound for the maximal entry of the principal eigenvector associated with the spectral radius of the adjacency matrix of a graph G. In 2002, Zhao and Hong [14] investigated bounds for maximal entry of the principal eigenvector of a symmetric nonnegative irreducible matrix with zero trace. In 2007, Cioabˇ a and Gregory [4] improved the bound of Papendieck and Recht [11] in terms of the maximum degree of the graph. Cioabˇ a [3] gave a necessary and sufficient condition for a graph to be bipartite in terms of an eigenvector corresponding to the largest eigenvalue of the adjacency matrix of the graph. In 2009, Das [5] obtained an upper bound for the maximal entry of the principal eigenvector of the signless Laplacian matrix of a graph and also considered the principal eigenvector of a symmetric nonnegative irreducible matrix. In 2011, Das [6] studied the bounds for the maximal and minimal entries of the principal eigenvector of the distance matrix of a graph. Motivated by the work of
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these previous authors, we present bounds for the maximal and the minimal entries of the principal eigenvector of the distance signless Laplacian matrix of a graph G. The distance spectral radius and the distance signless Laplacian spectral radius of G are denoted by ∂(G) and ∂ Q (G), respectively. Proposition 1.1. The following results hold: • [8] t ≤ ∂(G) ≤ T and 2t ≤ ∂ Q (G) ≤ 2T . Moreover, any of the equalities occurs if and only if G is transmission regular; • [13] T < ∂ Q ≤ 2T . As an immediate consequence, we have ∂ Q ≥ t + n − 1. In this paper we give an upper bound for the spectral radius of DQ(G) depending on the number of vertices n and the independence number α of the graph G. Let G denote the complement of the graph G and G1 ∨ G2 the join of the graphs G1 and G2 . Denote by Kn , the complete graph on n vertices. Let G(n, α) denotes the set of all connected graphs with n vertices and independence number α, and G(n, α, ∗) = G ∈ G(n, α) : G∼ = K α ∨ H, deg G (vk ) = deg G (vj ) for vj , vk ∈ V (H) , where r(H) (≥ 0) the degree of regularity of graph H. Let CI(n, α) be the graph K α ∨ Kn−α , where 1 ≤ α ≤ n − 1. Note that CI(n, α) ∈ G(n, α, ∗). We obtain the following theorem: Theorem 1.2. Let G ∈ G(n, α). Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). Then 1/p (∂ Q + α − 2n + 2)p minp y ≤ min , (n − α)αp + α(∂ Q + α − 2n + 2)p 1/p
Q p ∂ − 3α + 4 − n . (1) (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p Moreover, the equality holds if and only if G ∼ = K α ∨ H ∈ G(n, α, ∗) with r(H) ≥ n − 2α. This result is analogous to the following theorem, presented by Das in [6]: Theorem 1.3. (See [6], Theorem 4.1.) Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ of D(G). Then 1/p (∂ − n + α + 1)p minp y ≤ , (n − α)αp + α(∂ − n + α + 1)p 1/p
(∂ − 2α + 2)p , (2) (n − α)(∂ − 2α + 2)p + α(n − α)p
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where α is the independence number of G. Moreover, the equality holds in (2) if and only if G ∼ = CI(n, α) = K α ∨ Kn−α . Remark 1.4. In fact, we observe that the equality holds in (2), not only for the graph CI(n, α), but for a much wider class of graphs. More specifically, the same class of graphs that have been presented in Theorem 1.2. For example, it is easy to see that the equality holds for the graph K 3 ∨ C4 , that is not isomorphic to K 3 ∨ K4 . Also, the complete bipartite graph Kα,n−α , where α ≥ n − α is not isomorphic to CI(n, α), but it satisfies the equality of Theorem 4.1 in [6]. In addition we obtain bounds for the minimal entry of the principal eigenvector of D (G) of graphs. Finally, we give a lower bound and an upper bound for the maximal entry of the principal eigenvector of the distance signless Laplacian matrix DQ (G) of graph G. We state the lower and upper bound as follows: Q
Theorem 1.5. Let G a connected graph. Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then
y
maxp
>
(n − Δ)(∂ Q − T )p−1 (∂ Q − t) T p−1 + (n − Δ)(∂ Q − T )p−1 + (n − 2)(n − Δ − 1) T p−1
1/(p−1) , (3)
where Δ, T and t denote the maximum degree, the maximal transmission and the minimal transmission of G, respectively. Theorem 1.6. Let G be a connected graph. Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then
y maxp ≤
T p−1 Q p p−1 (∂ − T ) + T − (d − 1) (n − δ − 1) T p−1
1/p ,
(4)
where d, δ and T denote the diameter, the minimum degree and the maximal transmission of G, respectively. Moreover, the equality holds if and only if G ∼ = Kn or G is isomorphic to a regular graph of diameter 2. 2. On the minimal entry of the principal eigenvector of the distance signless Laplacian matrix In this section, we will discuss on the minimal entry of the principal eigenvector of the distance signless Laplacian matrix DQ (G).
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Proposition 2.1. Let G ∼ = K α ∨ H ∈ G(n, α). Then •
T r(vi ) =
n + α − 2, 2(n − 1) − α − deg H (vi ),
if vi ∈ V (K α ) if vi ∈ V (H),
where deg H (vi ) denotes the degree of vertex vi in H; • G ∈ G(n, α, ∗) if and only if the vertices of V (H) have the same transmission. In this case, denoting the transmissions of vertices in V (K α ) and vertices in V (H) by Tα and TH , respectively, we have Tα = n + α − 2 and TH = 2(n − 1) − α − r(H), where r denotes the degree regularity of graph H. Proof. If vi ∈ V (K α ), then T r(vi ) = 2(α − 1) + n − α = n + α − 2. For vi ∈ V (H), it follows that T r(vi ) = deg H (vi ) +2(n −α−deg H (vi ) −1) +α = 2(n −1) −α−deg H (vi ). 2 Remark 2.2. When more than one graph G ∼ = K α ∨ H ∈ G(n, α) is under consideration, then we write Tα (G) (and TH (G)) instead of Tα (and TH ). Similarly to [5] and [6], it is possible to obtain bounds for the minimal entry of the principal eigenvector of the distance signless Laplacian matrix of graph G. For this we need the following results. Theorem 2.3. If G ∈ G(n, α, ∗), then (i) ∂ Q (G) =
1 5n − 2r − 8 + (3n − 2α − 2r)2 − 4α(5n − 4α − 4r) + 12α2 , 2
(5)
(ii) the eigencomponents of the p-normalized principal eigenvector corresponding to the eigenvalue ∂ Q (G) are 1/p (∂ Q − 4n + 3α + 2r + 4)p yi = when vi ∈ V (K α ), (n − α)αp + α(∂ Q − 4n + 3α + 2r + 4)p 1/p Q p ∂ − 3α + 4 − n when vj ∈ V (H). and yj = (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
Proof. Since G ∈ G(n, α, ∗), its distance signless Laplacian matrix can be written as DQ (G) =
(2J + (n + α − 4)I)α J(n−α)×α
Jα×(n−α) , Hn−α
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where Hn−α is a square matrix of order n − α such that the diagonal entries are TH and the sum of each row is equal to 2TH − α (J is a matrix where every element is equal to one and I is an identity matrix). Let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). Then we have DQ (G) Y = ∂ Q Y. For vi , vk ∈ V (K α ),
∂ Q yi = Tα yi + 2
y +
v ∈V (K α ), =i
and ∂ Q yk = Tα yk + 2
y
v ∈V (H)
y +
v ∈V (K α ), =k
y .
v ∈V (H)
From the above, we get (∂ Q − Tα + 2)(yi − yk ) = 0, that is,
yi = yk
as ∂ Q ≥ 2n − 2 > n + α − 4 = Tα − 2.
Similarly, we can prove that yj = yk for vj , vk ∈ V (H). It follows that the T p-normalized principal eigenvector of DQ (G) is of the form Y = (a, a, . . . , a, b, b, . . . , b) , α
n−α
where a and b satisfy ∂ Q a = (n + 3α − 4)a + (n − α)b and ∂ Q b = αa + (2TH − α)b. Then a=
(n − α)b αa and b = Q . ∂ Q − n − 3α + 4 ∂ − 2TH + α
(6)
Eliminating a and b, we get the distance signless Laplacian spectral radius of a graph, ∂ Q (G) =
1 n − 4 + 2TH + 2α + (n − 4 − 2TH )2 + 4α(3n − 8 − 4TH ) + 12α2 , 2
which gives the required result in (i) as TH = 2n − 2 − r − α, concluding the first part of the proof. Since Y is the p-normalized principal eigenvector corresponding to the spectral radius Q ∂ of DQ (G), it follows that
1/p (∂ Q − 2TH + α)p a= , (n − α)αp + α(∂ Q + α − 2TH )p 1/p p Q ∂ − 3α + 4 − n b= . (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
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Again since TH = 2n − 2 − r − α, we get the required result in (ii). This completes the proof. 2 Note that any graph G ∈ G(n, α) can be obtained from CI(n, α) by removing edges. From this, using the Perron–Frobenius theorem, we obtain a new lower bound for ∂ Q (G), based on the number of vertices and the independence number of G. Theorem 2.4. If G ∈ G(n, α) then ∂ Q (G) ≥ ∂ Q (CI(n, α)) =
1 3n + 2α − 6 + (n + 2 − 2α)2 + 8α(α − 1) 2
with equality holding if and only if G ∼ = CI(n, α). Proof. Putting r = n − α − 1 in (5), we get the required result. 2 In fact, concerning the family G(n, α, ∗), we have the following proposition: Proposition 2.5. Let G ∼ = K α ∨ H ∈ G(n, α, ∗). Then the following statements are equivalent: (i) r ≥ n − 2α; (ii) TH ≤ Tα , that is, the minimum transmission of G is obtained to the vertices in V (H); T (iii) b ≤ a, where Y = (y1 , y2 , . . . , yn ) is the p-normalized principal eigenvector of DQ (G) and yi = a for vi ∈ V (K α ), yj = b for vj ∈ V (H). In this case, we have y
minp
=b=
1/p Q p ∂ − 3α + 4 − n . (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
Proof. (i) ⇐⇒ (ii): Since Tα = n + α − 2 and TH = 2n − r − α − 2, we have the required result. (ii) ⇐⇒ (iii): From (6), we have b ≤ a if and only if ∂ Q − 2TH + α ≥ α and ∂ Q − n −3α+4 ≤ n −α, which is equivalent to 2TH ≤ ∂ Q and ∂ Q ≤ 2(n +α−2) = 2Tα . 2 For a subset W of V (G), let G − W be the subgraph of G obtained by deleting the vertices of W and the edges incident to them. In Theorem 1.2, we give an upper bound for y minp , the minimal entry of the principal eigenvector of the distance signless Laplacian matrix, depending on the number of vertices, the spectral radius and the independence number of the graph. Although the first part of the proof of Theorem 1.2 is similar to the proof of the result in [6, Theorem 4.1], with adaptations related to the fact that the
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main diagonal, in this case, is non-zero, it becomes necessary to discuss the second part of the proof. Proof of Theorem 1.2. Since G has independence number α, there are two sets A and B such that V (G) = A ∪ B with A = {v1 , v2 , . . . , vα }, B = {vα+1 , vα+2 , . . . , vn } and there are no two adjacent vertices in the set A. Let vi ∈ A and vj ∈ B such that yi ≤ yk , ∀vk ∈ A, and yj ≤ yk , ∀vk ∈ B. Since DQ Y = ∂ Q Y, we obtain that α
∂ Q yi = Ti yi +
n
di, k yk +
k=1, k=i
di, k yk ≥ Ti yi +
k=α+1
α
di, k yi +
k=1, k=i
n
di, k yj .
k=α+1
As di,k ≥ 2 if vk ∈ A, and di,k ≥ 1 if vk ∈ B, it follows that α
di, k ≥ 2(α − 1) and
k=1, k=i
n
di, k ≥ n − α.
k=α+1
Therefore Ti ≥ 2(α − 1) + n − α = α + n − 2. Thus we have ∂ Q yi ≥ (2(α − 1) + α + n − 2)yi + (n − α)yj , that is, yi ≥
(n − α)yj . (∂ Q − 3α + 4 − n)
(7)
Similarly, we have ∂ Q yj ≥ Tj yj +
α
n
dj, k yi +
k=1
dj, k yj .
k=α+1, k=j
As dj,k ≥ 1∀k = j, it follows that α k=1
dj, k ≥ α,
n
dj, k ≥ n − α − 1 and Tj ≥ n − 1.
k=α+1,k=j
Then we have ∂ Q yj ≥ αyi + (2n − 2 − α)yj , that is, yj ≥
(∂ Q
αyi . + α − 2n + 2)
(8)
By the fact that the vector Y is p-normalized, we obtain α yi p + (n − α) yj p ≤
n k=1
ykp = 1.
(9)
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Substituting (8) in (9), we obtain
yi ≤
(∂ Q + α − 2n + 2)p α(∂ Q + α − 2n + 2)p + (n − α)αp
1/p (10)
and substituting (7) in (9), we obtain
yj ≤
(∂ Q − 3α + 4 − n)p Q (n − α)(∂ − 3α + 4 − n)p + α(n − α)p
1/p .
(11)
The first part of the proof is done. Suppose that equality holds in (1). Then all inequalities in the above must be equalities. From equality in (9), we get yk = yi for vk ∈ A and yk = yj for vk ∈ B. We consider the following two cases: Case (i): y minp = yi ≤ yj . Then we have yj =
αyj αyi ≤ Q , that is, ∂ Q ≤ 2n − 2. (∂ Q + α − 2n + 2) (∂ + α − 2n + 2)
From the above with Proposition 1.1, we get G ∼ = Kn . Case (ii): y minp = yj ≤ yi . Then we have yi =
(∂ Q
(n − α)yj , − 3α + 4 − n)
that is, ∂ Q yi = (n + α − 2) yi + (2α − 2) yi + (n − α) yj . For v ∈ A, we have α
∂ Q yi = T yi +
k=1, k=
n
d, k yi +
d, k yj .
k=α+1
From the above two results, we get (T − n − α + 2) yi +
α k=1, k=
(d, k − 2) yi +
n
(d, k − 1) yj = 0.
k=α+1
Since T ≥ n + α − 2, v ∈ A and d, k ≥ 1 for vk ∈ B, d, k ≥ 2 for vk ∈ A (k = ), we must have d, k = 1 for vk ∈ B, d, k = 2 for vk ∈ A (k = ), for any v ∈ A. Thus
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each vertex in A is adjacent to any vertex in B. Hence d = 2 and G ∼ = K α ∨ H, where H = G − A. We now show that any two vertices have the same transmission in B, that is, Tj = Ts for vs ∈ B. Indeed, for s such that α + 1 ≤ s ≤ n, remembering that ys = yj , ∀vs ∈ B, ∂ Q yj = Ts yj +
α
n
ds, k yi +
k=1
ds, k yj
k=α+1, k=s
= α yi + (2Ts − α) yj . Thus we have Ts = Tj for all vs ∈ B. Since d = 2, we have deg H (vs ) = deg H (vj ) for all vs ∈ B. Therefore H = G − A is a regular graph and hence G ∈ G(n, α, ∗). By Proposition 2.5, we have r(H) ≥ n − 2α. Conversely, if G ∼ = K α ∨ H ∈ G(n, α, ∗) with r ≥ n − 2α, then the equality follows directly from Proposition 2.5. 2 Noting that Tk ≥ t, ∀vk ∈ V (G), and ∂ Q ≥ 2t ≥ t + n − 1, by Proposition 1.1, we obtain another upper bound for y minp . Theorem 2.6. Let G ∈ G(n, α). Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). Then 1/p (∂ Q + α − n + 1 − t)p minp y ≤ min , α(∂ Q + α − n + 1 − t)p + (n − α)αp 1/p
(∂ Q − 2α + 2 − t)p . (12) (n − α)(∂ Q − 2α + 2 − t)p + α(n − α)p Moreover, the equality holds in (12) if and only if G ∈ G(n, α, ∗) is transmission regular. In this case G is (n + α − 2)-transmission regular with ∂ Q = 2(n + α − 2) and 1/p y minp = n1 . Remark 2.7. In order to reach the bound presented in the last theorem we have G ∼ = K α ∨ H ∈ G(n, α, ∗), and TH = Tα , that is, r(H) = n − 2α. A natural question is to ask, which theorem provides a better bound. To compare the two bounds for yi , we consider the function f (x) =
αxp
xp , + (n − α)αp
∀x ∈ (0, ∞).
Then f (x) =
(n − α) αp p xp−1 > 0, [αxp + (n − α)αp ]2
∀x ∈ (0, ∞),
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since p ≥ 1 and 0 < α < n. Therefore, the function f (x) is increasing for x > 0. This, together with the fact that ∂ Q + α − 2n + 2 ≥ ∂ Q + α − n + 1 − t ⇐⇒ t ≥ n − 1, which is always true. Therefore we conclude that
1/p (∂ Q + α − n + 1 − t)p yi ≤ α(∂ Q + α − n + 1 − t)p + (n − α)αp
1/p (∂ Q + α − 2n + 2)p ≤ , α(∂ Q + α − 2n + 2)p + (n − α)αp for any graph G. In order to analyze the bound for yj , we consider the function g(x) =
xp , (n − α)xp + α(n − α)p
∀x ∈ (0, ∞).
Then g (x) =
α (n − α)p p xp−1 [(n − α)xp + α(n − α)p ]
2
> 0,
∀x ∈ (0, ∞).
As ∂ Q − 2α + 2 − t ≥ ∂ Q − 3α + 4 − n ⇐⇒ t ≤ n + α − 2, using the fact that the function g(x) is increasing for x > 0, we have
yj ≤
≤
(∂ Q − 3α + 4 − n)p (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
1/p
(∂ Q − 2α + 2 − t)p Q (n − α)((∂ − 2α + 2 − t))p + α(n − α)p
1/p
if and only if the graph G has t ≤ n + α − 2. The next Corollary combines the results of the last two theorems in order to exhibit the bound as best as possible. Corollary 2.8. Let G ∈ G(n, α). Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). (i) If t ≥ n + α − 2, then y
minp
≤ min
(∂ Q + α − n + 1 − t)p Q α(∂ + α − n + 1 − t)p + (n − α)αp
(∂ Q − 2α + 2 − t)p (n − α)(∂ Q − 2α + 2 − t)p + α(n − α)p
1/p
1/p
,
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with equality holding if and only if G ∈ G(n, α, ∗) and, in this case, the graph G is 1/p (n + α − 2)-transmission regular with ∂ Q = 2(n + α − 2) and y minp = n1 . (ii) If t ≤ n + α − 2, then
y
minp
1/p (∂ Q + α − n + 1 − t)p , α(∂ Q + α − n + 1 − t)p + (n − α)αp 1/p
Q p ∂ − 3α + 4 − n (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
≤ min
with equality holding if and only if G ∼ = K α ∨ H ∈ G(n, α, ∗) and TH ≤ Tα , where we have y minp =
1/p p ∂ Q − 3α + 4 − n . (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
The next two examples exhibit families of graphs, showing that, in general t and n + α − 2 are incomparable. 2 Example 1. Let G ∼ = Pn . Then α = n2 and t = n4 . So ⎧ ⎨ t < n + α − 2 if n = 3 t = n + α − 2 if n = 2, 4, 5 ⎩ t > n + α − 2 if n ≥ 6. For G ∼ = Kp, q (complete bipartite graph) with p ≤ q, we have α = q and t = 2(p − 1) + q = n + p − 2. Therefore t ≤ n + α − 2. Since we are determining upper bound for the minimal entry of the principal eigenvector of the distance signless Laplacian matrix of a graph G, it becomes interesting to determine upper bound that do not depend on the value ∂ Q . In this sense, again utilizing the fact that the functions f (x) and g(x) are increasing and ∂ Q ≤ 2T with equality occurring if and only if the graph is transmission regular (Proposition 1.1), we have: Corollary 2.9. Let G ∈ G(n, α). Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the pnormalized principal eigenvector corresponding to ∂ Q . (i) If t ≥ n + α − 2, y
minp
≤ min
(2T + α − n + 1 − t)p α(2T + α − n + 1 − t)p + (n − α)αp
(2T − 2α + 2 − t)p (n − α)(2T − 2α + 2 − t)p + α(n − α)p
1/p
1/p
.
,
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(ii) If t ≤ n + α − 2, y
minp
≤
(2T + α − n + 1 − t)p α(2T + α − n + 1 − t)p + (n − α)αp
1/p
p
(2T − 3α + 4 − n) (n − α)(2T − 3α + 4 − n)p + α(n − α)p
, 1/p
.
In both cases the equality occurs if and only if G ∈ G(n, α) is transmission regular 1/p with t = n + α − 2, ∂ Q = 2(n + α − 2) and y minp = n1 . 3. On the largest entry of the principal eigenvector of the distance signless Laplacian matrix From now on, we turn our attention to obtain bounds for the maximal entry of the principal eigenvector, y maxp , associated to the spectral radius ∂ Q (G) of a connected graph G. We recall now a result that will be useful: Lemma 3.1. (See [5].) Let B = (bi,j )n×n be a symmetric nonnegative irreducible matrix with R, the largest diagonal entry and r, the smallest diagonal entry. Let Y = (y1 , y2 , . . . , yn )T be the p-norm normalized principle eigenvector of B corresponding to the spectral radius ρ and y1 ≥ y2 ≥ · · · ≥ yn . If p ≥ 2, then
y
maxp
(n − 1)(p−2)/2 (ρ − r)p/2 = y1 ≤ (n − 1)(p−2)/2 (ρ − r)p/2 + (ρ − R)p/2
1/p (13)
with equality holding in (13) if and only if B = c Ω, where c > 0 and the matrix Ω is defined as Ω = [ωi, j ], where
ωi, j
⎧ ⎪ R/c ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r/c ⎪ ⎪ ⎨ = 1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩0
if i = j = 1, if i = j ≥ 2, if i = 1, j ≥ 2, if j = 1, i ≥ 2, if i ≥ 2, j ≥ 2, i = j.
We now give an upper bound for the largest entry of the principal eigenvector of the distance signless Laplacian matrix of G. Corollary 3.2. Let G be a connected graph and also let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector associated to the spectral radius ∂ Q of DQ (G) with y1 ≥ y2 ≥ · · · ≥ yn . If p ≥ 2, then
K.Ch. Das et al. / Linear Algebra and its Applications 483 (2015) 200–220
y maxp = y1 ≤
(n − 1)(p−2)/2 (∂ Q − t)p/2 (p−2)/2 (n − 1) (∂ Q − t)p/2 + (∂ Q − T )p/2
213
1/p ,
(14)
where T and t denote, respectively, the maximal and minimal transmission of G. The equality occurs if and only if G ∼ = K2 . Proof. Setting R = T , r = t and ρ = ∂ Q in (13), we can get the required result in (14). The equality occurs in (14) if and only if there is a graph G such that its distance signless Laplacian matrix is written as DQ (G) = c Ω, where c > 0 and Ω is defined in Lemma 3.1. Since all the entries of the distance signless Laplacian matrix are greater than or equal to 1, it follows that G ∼ = K2 as G is connected graph and ωi, j = 0, i, j ≥ 2, i = j. Since c = r = R = 1, we have DQ (K2 ) = c Ω, the result is proven. 2 Remark 3.3. If p = 2 and the graph is transmission regular, the bound becomes y maxp ≤ √ 2 2 . Analogously to the result stated in [6] for distance matrix, we present, in the sequence, upper and lower bounds for the maximal entry of the principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). The lower bound is mentioned in Theorem 1.5. The proof is as follows: Proof of Theorem 1.5. Since Y is the p-normalized principal eigenvector associated to the spectral radius ∂ Q of DQ (G) and y1 ≥ y2 ≥ · · · ≥ yn , we have n
1=
ykp ≤ y1p + ynp + (n − 2) y2p ,
k=1
that is, ynp ≥ 1 − y1p − (n − 2) y2p .
(15)
Moreover, DQ Y = ∂ Q Y. For v1 ∈ V (G), ∂ Q y1 = T1 y1 +
n
d1, k yk ≥ t y1 +
k=2
n
yk + (n − deg G (v1 ) − 1) yn .
(16)
k=2
Since p ≥ 1, we multiply both sides of (16) by y2p−1 , we get ∂
Q
y1 y2p−1
≥
t y1 y2p−1
+
n
yk y2p−1 + (n − deg G (v1 ) − 1) yn y2p−1
k=2
≥ t y1 y2p−1 +
n k=2
ykp + (n − deg G (v1 ) − 1) ynp
(17)
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as y2 ≥ y3 ≥ · · · ≥ yn
" # ≥ t y2p−1 y1 + 1 − y1p + (n − deg G (v1 ) − 1) 1 − y1p − (n − 2) y2p ≥ t y2p−1 y1 + (n − Δ) (1 − y1p ) − (n − 2) (n − Δ − 1) y2p as deg G (v1 ) ≤ Δ ≥ t y2p−1 y1 + (n − Δ) (1 − y1p ) − (n − 2) (n − Δ − 1) y2p−1 as p ≥ 1. (18) Thus we have
y2 ≥
(n − Δ) (1 − y1p ) (∂ Q − t) y1 + (n − 2) (n − Δ − 1)
1/(p−1) .
(19)
For v2 ∈ V (G), we have n
∂ Q y2 = T2 y2 +
d2, k yk ≤ T y2 + T y1 .
(20)
k=1, k=2
From (19) and (20), we get (n − Δ) (1 − y1p ) (∂ Q − T )p−1 ≤ T p−1 y1p−1 , (∂ Q − t) y1 + (n − 2) (n − Δ − 1) that is, (n − Δ) (∂ Q − T )p−1 − (n − 2) (n − Δ − 1) T p−1 y1p−1 $ % ≤ (∂ Q − t) T p−1 + (n − Δ)(∂ Q − T )p−1 y1p $ % ≤ (∂ Q − t) T p−1 + (n − Δ)(∂ Q − T )p−1 y1p−1 ,
(21)
which gives the required result in (3). Now we have to prove that the inequality is strict. Since G is connected, yi > 0, i = 1, 2, . . . , n. Moreover, 2t ≤ ∂ Q ≤ 2T . Thus we have Q ∂ − t > 0, T > 0 and y1p < y1p−1 . Therefore one can see easily that (21) is strict. This completes the proof. 2 Corollary 3.4. Let G be a connected graph. Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then y maxp ≥
∂Q p−1
(2T )
1/p
p−1
(∂ Q − t) + ∂ Q
p−1
,
(22)
where T and t denote the maximal and minimal transmissions of G, respectively. Moreover, the equality holds in (22) if and only if G ∼ = Kn .
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215
Proof. From (17), we get ∂ Q y1 y2p−1 ≥ 1 − y1p + t y2p−1 y1 ,
(23)
that is,
1 − y1p y1 (∂ Q − t)
y2 ≥
1/p−1 .
(24)
From (20), we get ∂ Q y2 ≤ 2 T y1 .
(25)
From the above two results, we get
∂
Q
1 − y1p y1 (∂ Q − t)
1/p−1 ≤ 2 T y1 ,
implying y1p ≥
∂Q p−1
(2T )
p−1
(∂ Q − t) + ∂ Q
p−1 ,
which gives the required result in (22). The first part of the proof is done. Now suppose that equality holds in (22). Then all inequalities in the above must be equalities. In particular, from equality in (23), we get deg G (v1 ) = n − 1, T1 = t = n − 1 and y2 = y3 = · · · = yn . Again from equality in (25), we get T2 = T and y1 = y3 = y4 = · · · = yn . Thus we have y1 = y2 = · · · = yn . For vi ∈ V (G), ∂ Q = 2Ti , i = 1, 2, . . . , n; that is T1 = T2 = · · · = Tn and hence T = t = n − 1. Therefore deg G (v1 ) = deg G (v2 ) = · · · = deg G (vn ) = n − 1. Hence G ∼ = Kn . Conversely, let G ∼ = Kn . Then we have ∂ Q = 2(n − 1), T = t = n − 1 and y1 = y2 = 1 1/p . Hence the equality holds in (22) for Kn . 2 · · · = yn = n Analogously to what have been done for y minp , we also present in the next Corollary, bounds for y maxp do not depending on its largest eigenvalue. Corollary 3.5. Let G be a connected graph. Fixed p ≥ 3, let Y = (y1 , y2 , . . . , yn )T the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then
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y
maxp
≥
1/p
p−1
(2 t) p−1
(2 T )
,
p−1
t + (2 t)
where T and t denote the maximal and minimal transmissions of G, respectively. If p = 1 or p = 2, then y
maxp
≥
1/p
p−1
(2 T ) p−1
(2 T )
p−1
(2 T − t) + (2 T )
.
In any case, the equality occurs if and only if G ∼ = Kn . Proof. Fixed a > 0, consider the function f (x) =
xp−1 , a (x − t) + xp−1
2 t ≤ x ≤ 2 T.
Then we have f (x) =
a xp−2 ((p − 1) (x − t) − x) (a (x − t) + xp−1 )2
(26)
and f (x) = 0 if and only if x =
t (p − 1) . p−2
It is easy to see that f (x) ≥ 0 as x ≥ 2t ≥
t (p − 1) for p ≥ 3. p−2
If p = 1 or p = 2, then from (26), we get f (x) ≤ 0. Using these results together with the fact that 2t ≤ ∂ Q ≤ 2T and Corollary 3.4 we obtain the lower bound. Moreover, in any case, the equality holds if and only if G ∼ = Kn , by Corollary 3.4. 2 Lemma 3.6. Let G be an r-regular graph of diameter 2. Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then T1 = T2 = · · · = Tn = 2n − r − 2, ∂ Q (G) = 2(2n − r − 2)
1/p 1 . and y1 = y2 = · · · = yn = n
K.Ch. Das et al. / Linear Algebra and its Applications 483 (2015) 200–220
217
Proof. Since G is r-regular graph with d = 2, we have Ti = r + 2(n − 1 − r) = 2(n − 1) − r and hence T1 = T2 = · · · = Tn = T , (say). For vi ∈ V (G), n
∂ Q yi = Ti yi +
n
di, k yk , that is, (∂ Q − T ) yi =
k=1, k=i
di, k yk .
k=1, k=i
Taking summation from i = 1 to n, we get (∂ − T ) Q
n
yi =
i=1
n
n
di, k yk
i=1 k=1, k=i
=
n
[r + (n − 1 − r)2] yi
i=1
= (2n − 2 − r)
n
yi .
i=1
Since n
yi > 0,
i=1
from the above, we get ∂ Q (G) = T + 2n − r − 2. The first part of the proof is done. For v1 ∈ V (G), ∂ Q y1 = T1 y1 +
n
d1, k yk ≤ T y1 + (2n − 2 − r) y2 .
(27)
k=2
Since ∂ Q (G) = T + 2n − r − 2, from the above, we get y1 ≤ y2 . But we have y1 ≥ y2 . Thus we have y1 = y2 and hence from (27), y1 = y2 = · · · = yn . Since n
yip = 1,
i=1
we have y1 = y2 = · · · = yn =
1 1/p n
.
2
In Theorem 1.6, we mentioned an upper bound for the maximal entry of the principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). The proof is as follows:
K.Ch. Das et al. / Linear Algebra and its Applications 483 (2015) 200–220
218
Proof of Theorem 1.6. For vi ∈ V (G), n
∂ Q yi = Ti yi +
di, k yk ,
k=1, k=i
that is, (∂ Q − T )yi ≤ (∂ Q − Ti )yi =
n
di, k yk .
(28)
k=1, k=i
Since p ≥ 1, by weighted power mean inequality, we get ⎛
n
⎜ k=1, k=i ⎜ n ⎝
⎛
⎞1/p
di, k ykp
di, k yk
⎜ k=1, k=i ≥⎜ n ⎝
⎟ ⎟ ⎠
di, k
⎞
n
k=1, k=i
di, k
⎟ ⎟, ⎠
k=1, k=i
that is, ⎛
⎞p
n
⎝
⎛
di, k yk ⎠ ≤ ⎝
k=1, k=i
⎞p−1
n
di, k ⎠
k=1, k=i
n
di, k ykp .
(29)
k=1, k=i
Since n
ykp = 1,
k=1
from (28) and (29), we get (∂ − T ) Q
p
yip
≤
Tip−1
(1 −
yip )
+
n
(di, k −
1) ykp
.
(30)
k=1, k=i
Setting i = 1 in (30), we get (∂ − T ) Q
p
y1p
≤T
p−1
≤T
p−1
(1 −
y1p )
+ (d − 1) (n − deg G (v1 ) −
1) y2p
as T ≥ T1
(1 −
y1p )
+ (d − 1) (n − δ −
1) y2p
.
(31)
Similarly, setting i = 2 in (30), we get (∂ Q −
T )p y2p
≤ T p−1 (1 −
y2p )
+ (d − 1) (n − δ −
1) y1p
,
(32)
K.Ch. Das et al. / Linear Algebra and its Applications 483 (2015) 200–220
219
that is, y2p ≤
(d − 1) (n − δ − 1) y1p + 1 p−1 T . (∂ Q − T )p + T p−1
Putting this value in (31), we get
(∂ − T ) + T Q
p
p−1
+ (d − 1) (n − δ − 1) T
y1p ≤
(∂ Q
T p−1
2
(∂ Q − T )p + T p−1 =
p−1
−
T )p
+
T p−1
− (d − 1)2 (n − δ − 1)2 T 2p−2
T p−1 , − (d − 1) (n − δ − 1) T p−1
which gives the required result in (4). The first part of the proof is done. Now suppose that equality holds in (4). Then all inequalities in the above must be equalities. In particular, from equality in (28), we get Ti = T . From equality in (29), we get y1 = y2 = · · · = yi−1 = yi+1 = · · · = yn . From equality in (31), we get deg G (v1 ) = δ, T1 = T , y2 = y3 = · · · = yn and d ≤ 2. From equality in (32), we get deg G (v2 ) = δ, T2 = T , y1 = y3 = · · · = yn and d ≤ 2. Thus we have y1 = y2 = · · · = yn = y, (say). If d = 1, G ∼ = Kn . Otherwise, d = 2. For vi ∈ V (G), " # ∂ Q y = 2Ti y, that is, ∂ Q = 2 deg G (vi ) + 2(n − 1 − deg G (vi )) = 4(n − 1) − 2 deg G (vi ). From the above, we conclude that deg G (v1 ) = deg G (v2 ) = · · · = deg G (vn ). Hence G is a regular graph of diameter 2. Conversely, let G ∼ = Kn . Then we have ∂ Q = 2(n − 1), T = n − 1 = δ, d = 1 and 1/p y1 = y2 = · · · = yn = n1 . One can see easily that equality holds in (4). Let G be an r-regular graph of diameter 2. Then by Lemma 3.6, we get d = 2, 1/p T1 = T2 = · · · = Tn = 2n − r − 2 = T , (say), y1 = y2 = · · · = yn = n1 and ∂ Q (G) = 2T . Then T p−1 T p−1 = (∂ Q − T )p + T p−1 − (d − 1)(n − δ − 1) T p−1 T p + T p−1 − (n − r − 1) T p−1 1 T −n+r+2 1 = = y1p as T = 2n − r − 2. n
=
Hence equality holds in (4). 2
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References [1] M. Aouchiche, P. Hansen, Distance spectra of graphs: a survey, Linear Algebra Appl. 458 (2014) 301–386. [2] M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013) 21–33. a, A necessary and sufficient eigenvector condition for a graph to be bipartite, Electron. [3] S.M. Cioabˇ J. Linear Algebra 20 (2010) 351–353. a, D.A. Gregory, Principal eigenvectors of irregular graphs, Electron. J. Linear Algebra [4] S.M. Cioabˇ 16 (2007) 366–379. [5] K.C. Das, A sharp upper bound on the maximal entry in the principal eigenvector of symmetric nonnegative matrix, Linear Algebra Appl. 431 (2009) 1340–1350. [6] K.C. Das, Maximal and minimal entry in the principal eigenvector for the distance matrix of a graph, Discrete Math. 311 (2011) 2593–2600. [7] K.C. Das, Proof of conjectures on the distance signless Laplacian eigenvalues of graphs, Linear Algebra Appl. 467 (2015) 100–115. [8] W. Hong, L. You, Further results on the spectral radius of matrices and graphs, Appl. Math. Comput. 239 (2014) 326–332. [9] R.A. Horn, C.R. Johnson, Matrix Analysis, second edition, Cambridge University Press, 2012. [10] X. Li, Y. Fan, S. Zha, A lower bound for the distance signless Laplacian spectral radius of graphs in terms of chromatic number, J. Math. Res. Appl. 14 (3) (2014) 289–294. [11] B. Papendieck, P. Recht, On maximal entries in the principal eigenvector of graphs, Linear Algebra Appl. 310 (2000) 129–138. [12] R. Xing, B. Zhou, On the distance and distance signless Laplacian spectral radii of bicyclic graphs, Linear Algebra Appl. 439 (2013) 3955–3963. [13] R. Xing, B. Zhou, J. Lia, On the distance signless Laplacian spectral radius of graphs, Linear Multilinear Algebra 62 (2014) 1377–1387. [14] S. Zhao, Y. Hong, On the bounds of maximal entries in the principal eigenvector of symmetric nonnegative matrix, Linear Algebra Appl. 340 (2002) 245–252.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Bounds on the entries of the principal eigenvector of the distance signless Laplacian matrix Kinkar Ch. Das a,∗,1 , Celso M. da Silva Junior b,2 , Maria Aguieiras A. de Freitas c,2 , Renata R. Del-Vecchio d,2 a
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea b PEP-COPPE, Universidade Federal do Rio de Janeiro and Centro Federal de Educação Tecnológica, Rio de Janeiro, Brazil c Instituto de Matemática and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil d Instituto de Matemática, Universidade Federal Fluminense, Rio de Janeiro, Brazil
a r t i c l e
i n f o
Article history: Received 31 October 2014 Accepted 2 June 2015 Available online xxxx Submitted by D. Stevanovic MSC: 05C12 05C50 15A18 Keywords: Distance signless Laplacian matrix Spectral radius Principal eigenvector Independence number Diameter
a b s t r a c t The distance signless Laplacian spectral radius of a connected graph G is the largest eigenvalue of the distance signless Laplacian matrix of G, defined as DQ (G) = T r(G) + D(G), where D(G) is the distance matrix of G and T r(G) is the diagonal matrix of vertex transmissions of G. In this paper we determine upper and lower bounds on the minimal and maximal entries of the principal eigenvector of DQ (G) and characterize the extremal graphs. In addition, we obtain a lower bound on the distance signless Laplacian spectral radius of G based on its order and independence number, and characterize the extremal graph. © 2015 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: [email protected] (K.Ch. Das), [email protected] (C.M. da Silva Junior), [email protected] (M.A.A. de Freitas), [email protected]ff.br (R.R. Del-Vecchio). 1 Work supported by National Research Foundation of Korea with the grant no. 2013R1A1A2009341. 2 Work partially supported by CNPq and FAPERJ-Brasil. http://dx.doi.org/10.1016/j.laa.2015.06.003 0024-3795/© 2015 Elsevier Inc. All rights reserved.
K.Ch. Das et al. / Linear Algebra and its Applications 483 (2015) 200–220
201
1. Introduction All graphs considered in this paper are finite, simple and connected. Let G = (V, E) be a graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G), where |V (G)| = n and |E(G)| = m. Let deg G (vi ) denote the degree of vertex vi in G. The maximum vertex degree is denoted by Δ and the minimum by δ. The graph G is called degree regular, or simply regular, if all vertices of G have the same degree. Let the distance (the length of a shortest path) between vertices vi and vj of G denoted by di, j . Denote by d the diameter of graph G. Given a graph G, a subset S of V (G) is said to be an independent set of G if the subgraph G[S], induced by S, is a graph with |S| isolated vertices. The independence number α = α(G) of G is the number of vertices in the largest independent set of G. The transmission T r(vi ) or simply Ti , of a vertex vi is defined as the sum of the distances from vi to all other vertices in G, i.e., T r(vi ) = di, j . We denote by T and t the vj ∈V (G)
maximal and minimal transmissions of G, respectively, that is, T = max 1≤i≤n T r(vi ) and t = min 1≤i≤n T r(vi ). The graph G is called k-transmission regular if T r(vi ) = k for every vertex vi ∈ V (G). The distance matrix of G, denoted by D(G), is the n × n matrix whose (i, j)-th entry is equal to di, j , i, j = 1, 2, . . . , n. For more details about the distance matrix we recommend [1]. In [2], Aouchiche and Hansen introduced the distance signless Laplacian matrix of a connected graph G as DQ (G) = T r(G) + D(G), where T r(G) is the diagonal matrix of vertex transmissions of G. This matrix has been studied by several researchers and recent results can be found in [7,8,10,12,13]. For a connected graph G the adjacency, signless Laplacian, distance and distance signless Laplacian matrices are symmetric, nonnegative and irreducible. Therefore, by the Perron–Frobenius theorem [9], fixed p (1 ≤ p < ∞), each of these matrices has a unique eigenvector Y = (y1 , y2 , . . . , yn )T , positive and unitary (Yp = 1) associated with its spectral radius. This eigenvector is called the p-normalized principal eigenvector of the matrix. The study of the principal eigenvector has aroused interest, as evidenced by the articles mentioned below. In 2000, Papendieck and Recht [11] obtained an upper bound for the maximal entry of the principal eigenvector associated with the spectral radius of the adjacency matrix of a graph G. In 2002, Zhao and Hong [14] investigated bounds for maximal entry of the principal eigenvector of a symmetric nonnegative irreducible matrix with zero trace. In 2007, Cioabˇ a and Gregory [4] improved the bound of Papendieck and Recht [11] in terms of the maximum degree of the graph. Cioabˇ a [3] gave a necessary and sufficient condition for a graph to be bipartite in terms of an eigenvector corresponding to the largest eigenvalue of the adjacency matrix of the graph. In 2009, Das [5] obtained an upper bound for the maximal entry of the principal eigenvector of the signless Laplacian matrix of a graph and also considered the principal eigenvector of a symmetric nonnegative irreducible matrix. In 2011, Das [6] studied the bounds for the maximal and minimal entries of the principal eigenvector of the distance matrix of a graph. Motivated by the work of
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these previous authors, we present bounds for the maximal and the minimal entries of the principal eigenvector of the distance signless Laplacian matrix of a graph G. The distance spectral radius and the distance signless Laplacian spectral radius of G are denoted by ∂(G) and ∂ Q (G), respectively. Proposition 1.1. The following results hold: • [8] t ≤ ∂(G) ≤ T and 2t ≤ ∂ Q (G) ≤ 2T . Moreover, any of the equalities occurs if and only if G is transmission regular; • [13] T < ∂ Q ≤ 2T . As an immediate consequence, we have ∂ Q ≥ t + n − 1. In this paper we give an upper bound for the spectral radius of DQ(G) depending on the number of vertices n and the independence number α of the graph G. Let G denote the complement of the graph G and G1 ∨ G2 the join of the graphs G1 and G2 . Denote by Kn , the complete graph on n vertices. Let G(n, α) denotes the set of all connected graphs with n vertices and independence number α, and G(n, α, ∗) = G ∈ G(n, α) : G∼ = K α ∨ H, deg G (vk ) = deg G (vj ) for vj , vk ∈ V (H) , where r(H) (≥ 0) the degree of regularity of graph H. Let CI(n, α) be the graph K α ∨ Kn−α , where 1 ≤ α ≤ n − 1. Note that CI(n, α) ∈ G(n, α, ∗). We obtain the following theorem: Theorem 1.2. Let G ∈ G(n, α). Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). Then 1/p (∂ Q + α − 2n + 2)p minp y ≤ min , (n − α)αp + α(∂ Q + α − 2n + 2)p 1/p
Q p ∂ − 3α + 4 − n . (1) (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p Moreover, the equality holds if and only if G ∼ = K α ∨ H ∈ G(n, α, ∗) with r(H) ≥ n − 2α. This result is analogous to the following theorem, presented by Das in [6]: Theorem 1.3. (See [6], Theorem 4.1.) Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ of D(G). Then 1/p (∂ − n + α + 1)p minp y ≤ , (n − α)αp + α(∂ − n + α + 1)p 1/p
(∂ − 2α + 2)p , (2) (n − α)(∂ − 2α + 2)p + α(n − α)p
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where α is the independence number of G. Moreover, the equality holds in (2) if and only if G ∼ = CI(n, α) = K α ∨ Kn−α . Remark 1.4. In fact, we observe that the equality holds in (2), not only for the graph CI(n, α), but for a much wider class of graphs. More specifically, the same class of graphs that have been presented in Theorem 1.2. For example, it is easy to see that the equality holds for the graph K 3 ∨ C4 , that is not isomorphic to K 3 ∨ K4 . Also, the complete bipartite graph Kα,n−α , where α ≥ n − α is not isomorphic to CI(n, α), but it satisfies the equality of Theorem 4.1 in [6]. In addition we obtain bounds for the minimal entry of the principal eigenvector of D (G) of graphs. Finally, we give a lower bound and an upper bound for the maximal entry of the principal eigenvector of the distance signless Laplacian matrix DQ (G) of graph G. We state the lower and upper bound as follows: Q
Theorem 1.5. Let G a connected graph. Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then
y
maxp
>
(n − Δ)(∂ Q − T )p−1 (∂ Q − t) T p−1 + (n − Δ)(∂ Q − T )p−1 + (n − 2)(n − Δ − 1) T p−1
1/(p−1) , (3)
where Δ, T and t denote the maximum degree, the maximal transmission and the minimal transmission of G, respectively. Theorem 1.6. Let G be a connected graph. Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then
y maxp ≤
T p−1 Q p p−1 (∂ − T ) + T − (d − 1) (n − δ − 1) T p−1
1/p ,
(4)
where d, δ and T denote the diameter, the minimum degree and the maximal transmission of G, respectively. Moreover, the equality holds if and only if G ∼ = Kn or G is isomorphic to a regular graph of diameter 2. 2. On the minimal entry of the principal eigenvector of the distance signless Laplacian matrix In this section, we will discuss on the minimal entry of the principal eigenvector of the distance signless Laplacian matrix DQ (G).
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Proposition 2.1. Let G ∼ = K α ∨ H ∈ G(n, α). Then •
T r(vi ) =
n + α − 2, 2(n − 1) − α − deg H (vi ),
if vi ∈ V (K α ) if vi ∈ V (H),
where deg H (vi ) denotes the degree of vertex vi in H; • G ∈ G(n, α, ∗) if and only if the vertices of V (H) have the same transmission. In this case, denoting the transmissions of vertices in V (K α ) and vertices in V (H) by Tα and TH , respectively, we have Tα = n + α − 2 and TH = 2(n − 1) − α − r(H), where r denotes the degree regularity of graph H. Proof. If vi ∈ V (K α ), then T r(vi ) = 2(α − 1) + n − α = n + α − 2. For vi ∈ V (H), it follows that T r(vi ) = deg H (vi ) +2(n −α−deg H (vi ) −1) +α = 2(n −1) −α−deg H (vi ). 2 Remark 2.2. When more than one graph G ∼ = K α ∨ H ∈ G(n, α) is under consideration, then we write Tα (G) (and TH (G)) instead of Tα (and TH ). Similarly to [5] and [6], it is possible to obtain bounds for the minimal entry of the principal eigenvector of the distance signless Laplacian matrix of graph G. For this we need the following results. Theorem 2.3. If G ∈ G(n, α, ∗), then (i) ∂ Q (G) =
1 5n − 2r − 8 + (3n − 2α − 2r)2 − 4α(5n − 4α − 4r) + 12α2 , 2
(5)
(ii) the eigencomponents of the p-normalized principal eigenvector corresponding to the eigenvalue ∂ Q (G) are 1/p (∂ Q − 4n + 3α + 2r + 4)p yi = when vi ∈ V (K α ), (n − α)αp + α(∂ Q − 4n + 3α + 2r + 4)p 1/p Q p ∂ − 3α + 4 − n when vj ∈ V (H). and yj = (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
Proof. Since G ∈ G(n, α, ∗), its distance signless Laplacian matrix can be written as DQ (G) =
(2J + (n + α − 4)I)α J(n−α)×α
Jα×(n−α) , Hn−α
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where Hn−α is a square matrix of order n − α such that the diagonal entries are TH and the sum of each row is equal to 2TH − α (J is a matrix where every element is equal to one and I is an identity matrix). Let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). Then we have DQ (G) Y = ∂ Q Y. For vi , vk ∈ V (K α ),
∂ Q yi = Tα yi + 2
y +
v ∈V (K α ), =i
and ∂ Q yk = Tα yk + 2
y
v ∈V (H)
y +
v ∈V (K α ), =k
y .
v ∈V (H)
From the above, we get (∂ Q − Tα + 2)(yi − yk ) = 0, that is,
yi = yk
as ∂ Q ≥ 2n − 2 > n + α − 4 = Tα − 2.
Similarly, we can prove that yj = yk for vj , vk ∈ V (H). It follows that the T p-normalized principal eigenvector of DQ (G) is of the form Y = (a, a, . . . , a, b, b, . . . , b) , α
n−α
where a and b satisfy ∂ Q a = (n + 3α − 4)a + (n − α)b and ∂ Q b = αa + (2TH − α)b. Then a=
(n − α)b αa and b = Q . ∂ Q − n − 3α + 4 ∂ − 2TH + α
(6)
Eliminating a and b, we get the distance signless Laplacian spectral radius of a graph, ∂ Q (G) =
1 n − 4 + 2TH + 2α + (n − 4 − 2TH )2 + 4α(3n − 8 − 4TH ) + 12α2 , 2
which gives the required result in (i) as TH = 2n − 2 − r − α, concluding the first part of the proof. Since Y is the p-normalized principal eigenvector corresponding to the spectral radius Q ∂ of DQ (G), it follows that
1/p (∂ Q − 2TH + α)p a= , (n − α)αp + α(∂ Q + α − 2TH )p 1/p p Q ∂ − 3α + 4 − n b= . (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
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Again since TH = 2n − 2 − r − α, we get the required result in (ii). This completes the proof. 2 Note that any graph G ∈ G(n, α) can be obtained from CI(n, α) by removing edges. From this, using the Perron–Frobenius theorem, we obtain a new lower bound for ∂ Q (G), based on the number of vertices and the independence number of G. Theorem 2.4. If G ∈ G(n, α) then ∂ Q (G) ≥ ∂ Q (CI(n, α)) =
1 3n + 2α − 6 + (n + 2 − 2α)2 + 8α(α − 1) 2
with equality holding if and only if G ∼ = CI(n, α). Proof. Putting r = n − α − 1 in (5), we get the required result. 2 In fact, concerning the family G(n, α, ∗), we have the following proposition: Proposition 2.5. Let G ∼ = K α ∨ H ∈ G(n, α, ∗). Then the following statements are equivalent: (i) r ≥ n − 2α; (ii) TH ≤ Tα , that is, the minimum transmission of G is obtained to the vertices in V (H); T (iii) b ≤ a, where Y = (y1 , y2 , . . . , yn ) is the p-normalized principal eigenvector of DQ (G) and yi = a for vi ∈ V (K α ), yj = b for vj ∈ V (H). In this case, we have y
minp
=b=
1/p Q p ∂ − 3α + 4 − n . (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
Proof. (i) ⇐⇒ (ii): Since Tα = n + α − 2 and TH = 2n − r − α − 2, we have the required result. (ii) ⇐⇒ (iii): From (6), we have b ≤ a if and only if ∂ Q − 2TH + α ≥ α and ∂ Q − n −3α+4 ≤ n −α, which is equivalent to 2TH ≤ ∂ Q and ∂ Q ≤ 2(n +α−2) = 2Tα . 2 For a subset W of V (G), let G − W be the subgraph of G obtained by deleting the vertices of W and the edges incident to them. In Theorem 1.2, we give an upper bound for y minp , the minimal entry of the principal eigenvector of the distance signless Laplacian matrix, depending on the number of vertices, the spectral radius and the independence number of the graph. Although the first part of the proof of Theorem 1.2 is similar to the proof of the result in [6, Theorem 4.1], with adaptations related to the fact that the
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main diagonal, in this case, is non-zero, it becomes necessary to discuss the second part of the proof. Proof of Theorem 1.2. Since G has independence number α, there are two sets A and B such that V (G) = A ∪ B with A = {v1 , v2 , . . . , vα }, B = {vα+1 , vα+2 , . . . , vn } and there are no two adjacent vertices in the set A. Let vi ∈ A and vj ∈ B such that yi ≤ yk , ∀vk ∈ A, and yj ≤ yk , ∀vk ∈ B. Since DQ Y = ∂ Q Y, we obtain that α
∂ Q yi = Ti yi +
n
di, k yk +
k=1, k=i
di, k yk ≥ Ti yi +
k=α+1
α
di, k yi +
k=1, k=i
n
di, k yj .
k=α+1
As di,k ≥ 2 if vk ∈ A, and di,k ≥ 1 if vk ∈ B, it follows that α
di, k ≥ 2(α − 1) and
k=1, k=i
n
di, k ≥ n − α.
k=α+1
Therefore Ti ≥ 2(α − 1) + n − α = α + n − 2. Thus we have ∂ Q yi ≥ (2(α − 1) + α + n − 2)yi + (n − α)yj , that is, yi ≥
(n − α)yj . (∂ Q − 3α + 4 − n)
(7)
Similarly, we have ∂ Q yj ≥ Tj yj +
α
n
dj, k yi +
k=1
dj, k yj .
k=α+1, k=j
As dj,k ≥ 1∀k = j, it follows that α k=1
dj, k ≥ α,
n
dj, k ≥ n − α − 1 and Tj ≥ n − 1.
k=α+1,k=j
Then we have ∂ Q yj ≥ αyi + (2n − 2 − α)yj , that is, yj ≥
(∂ Q
αyi . + α − 2n + 2)
(8)
By the fact that the vector Y is p-normalized, we obtain α yi p + (n − α) yj p ≤
n k=1
ykp = 1.
(9)
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Substituting (8) in (9), we obtain
yi ≤
(∂ Q + α − 2n + 2)p α(∂ Q + α − 2n + 2)p + (n − α)αp
1/p (10)
and substituting (7) in (9), we obtain
yj ≤
(∂ Q − 3α + 4 − n)p Q (n − α)(∂ − 3α + 4 − n)p + α(n − α)p
1/p .
(11)
The first part of the proof is done. Suppose that equality holds in (1). Then all inequalities in the above must be equalities. From equality in (9), we get yk = yi for vk ∈ A and yk = yj for vk ∈ B. We consider the following two cases: Case (i): y minp = yi ≤ yj . Then we have yj =
αyj αyi ≤ Q , that is, ∂ Q ≤ 2n − 2. (∂ Q + α − 2n + 2) (∂ + α − 2n + 2)
From the above with Proposition 1.1, we get G ∼ = Kn . Case (ii): y minp = yj ≤ yi . Then we have yi =
(∂ Q
(n − α)yj , − 3α + 4 − n)
that is, ∂ Q yi = (n + α − 2) yi + (2α − 2) yi + (n − α) yj . For v ∈ A, we have α
∂ Q yi = T yi +
k=1, k=
n
d, k yi +
d, k yj .
k=α+1
From the above two results, we get (T − n − α + 2) yi +
α k=1, k=
(d, k − 2) yi +
n
(d, k − 1) yj = 0.
k=α+1
Since T ≥ n + α − 2, v ∈ A and d, k ≥ 1 for vk ∈ B, d, k ≥ 2 for vk ∈ A (k = ), we must have d, k = 1 for vk ∈ B, d, k = 2 for vk ∈ A (k = ), for any v ∈ A. Thus
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each vertex in A is adjacent to any vertex in B. Hence d = 2 and G ∼ = K α ∨ H, where H = G − A. We now show that any two vertices have the same transmission in B, that is, Tj = Ts for vs ∈ B. Indeed, for s such that α + 1 ≤ s ≤ n, remembering that ys = yj , ∀vs ∈ B, ∂ Q yj = Ts yj +
α
n
ds, k yi +
k=1
ds, k yj
k=α+1, k=s
= α yi + (2Ts − α) yj . Thus we have Ts = Tj for all vs ∈ B. Since d = 2, we have deg H (vs ) = deg H (vj ) for all vs ∈ B. Therefore H = G − A is a regular graph and hence G ∈ G(n, α, ∗). By Proposition 2.5, we have r(H) ≥ n − 2α. Conversely, if G ∼ = K α ∨ H ∈ G(n, α, ∗) with r ≥ n − 2α, then the equality follows directly from Proposition 2.5. 2 Noting that Tk ≥ t, ∀vk ∈ V (G), and ∂ Q ≥ 2t ≥ t + n − 1, by Proposition 1.1, we obtain another upper bound for y minp . Theorem 2.6. Let G ∈ G(n, α). Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). Then 1/p (∂ Q + α − n + 1 − t)p minp y ≤ min , α(∂ Q + α − n + 1 − t)p + (n − α)αp 1/p
(∂ Q − 2α + 2 − t)p . (12) (n − α)(∂ Q − 2α + 2 − t)p + α(n − α)p Moreover, the equality holds in (12) if and only if G ∈ G(n, α, ∗) is transmission regular. In this case G is (n + α − 2)-transmission regular with ∂ Q = 2(n + α − 2) and 1/p y minp = n1 . Remark 2.7. In order to reach the bound presented in the last theorem we have G ∼ = K α ∨ H ∈ G(n, α, ∗), and TH = Tα , that is, r(H) = n − 2α. A natural question is to ask, which theorem provides a better bound. To compare the two bounds for yi , we consider the function f (x) =
αxp
xp , + (n − α)αp
∀x ∈ (0, ∞).
Then f (x) =
(n − α) αp p xp−1 > 0, [αxp + (n − α)αp ]2
∀x ∈ (0, ∞),
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since p ≥ 1 and 0 < α < n. Therefore, the function f (x) is increasing for x > 0. This, together with the fact that ∂ Q + α − 2n + 2 ≥ ∂ Q + α − n + 1 − t ⇐⇒ t ≥ n − 1, which is always true. Therefore we conclude that
1/p (∂ Q + α − n + 1 − t)p yi ≤ α(∂ Q + α − n + 1 − t)p + (n − α)αp
1/p (∂ Q + α − 2n + 2)p ≤ , α(∂ Q + α − 2n + 2)p + (n − α)αp for any graph G. In order to analyze the bound for yj , we consider the function g(x) =
xp , (n − α)xp + α(n − α)p
∀x ∈ (0, ∞).
Then g (x) =
α (n − α)p p xp−1 [(n − α)xp + α(n − α)p ]
2
> 0,
∀x ∈ (0, ∞).
As ∂ Q − 2α + 2 − t ≥ ∂ Q − 3α + 4 − n ⇐⇒ t ≤ n + α − 2, using the fact that the function g(x) is increasing for x > 0, we have
yj ≤
≤
(∂ Q − 3α + 4 − n)p (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
1/p
(∂ Q − 2α + 2 − t)p Q (n − α)((∂ − 2α + 2 − t))p + α(n − α)p
1/p
if and only if the graph G has t ≤ n + α − 2. The next Corollary combines the results of the last two theorems in order to exhibit the bound as best as possible. Corollary 2.8. Let G ∈ G(n, α). Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). (i) If t ≥ n + α − 2, then y
minp
≤ min
(∂ Q + α − n + 1 − t)p Q α(∂ + α − n + 1 − t)p + (n − α)αp
(∂ Q − 2α + 2 − t)p (n − α)(∂ Q − 2α + 2 − t)p + α(n − α)p
1/p
1/p
,
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211
with equality holding if and only if G ∈ G(n, α, ∗) and, in this case, the graph G is 1/p (n + α − 2)-transmission regular with ∂ Q = 2(n + α − 2) and y minp = n1 . (ii) If t ≤ n + α − 2, then
y
minp
1/p (∂ Q + α − n + 1 − t)p , α(∂ Q + α − n + 1 − t)p + (n − α)αp 1/p
Q p ∂ − 3α + 4 − n (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
≤ min
with equality holding if and only if G ∼ = K α ∨ H ∈ G(n, α, ∗) and TH ≤ Tα , where we have y minp =
1/p p ∂ Q − 3α + 4 − n . (n − α)(∂ Q − 3α + 4 − n)p + α(n − α)p
The next two examples exhibit families of graphs, showing that, in general t and n + α − 2 are incomparable. 2 Example 1. Let G ∼ = Pn . Then α = n2 and t = n4 . So ⎧ ⎨ t < n + α − 2 if n = 3 t = n + α − 2 if n = 2, 4, 5 ⎩ t > n + α − 2 if n ≥ 6. For G ∼ = Kp, q (complete bipartite graph) with p ≤ q, we have α = q and t = 2(p − 1) + q = n + p − 2. Therefore t ≤ n + α − 2. Since we are determining upper bound for the minimal entry of the principal eigenvector of the distance signless Laplacian matrix of a graph G, it becomes interesting to determine upper bound that do not depend on the value ∂ Q . In this sense, again utilizing the fact that the functions f (x) and g(x) are increasing and ∂ Q ≤ 2T with equality occurring if and only if the graph is transmission regular (Proposition 1.1), we have: Corollary 2.9. Let G ∈ G(n, α). Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the pnormalized principal eigenvector corresponding to ∂ Q . (i) If t ≥ n + α − 2, y
minp
≤ min
(2T + α − n + 1 − t)p α(2T + α − n + 1 − t)p + (n − α)αp
(2T − 2α + 2 − t)p (n − α)(2T − 2α + 2 − t)p + α(n − α)p
1/p
1/p
.
,
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(ii) If t ≤ n + α − 2, y
minp
≤
(2T + α − n + 1 − t)p α(2T + α − n + 1 − t)p + (n − α)αp
1/p
p
(2T − 3α + 4 − n) (n − α)(2T − 3α + 4 − n)p + α(n − α)p
, 1/p
.
In both cases the equality occurs if and only if G ∈ G(n, α) is transmission regular 1/p with t = n + α − 2, ∂ Q = 2(n + α − 2) and y minp = n1 . 3. On the largest entry of the principal eigenvector of the distance signless Laplacian matrix From now on, we turn our attention to obtain bounds for the maximal entry of the principal eigenvector, y maxp , associated to the spectral radius ∂ Q (G) of a connected graph G. We recall now a result that will be useful: Lemma 3.1. (See [5].) Let B = (bi,j )n×n be a symmetric nonnegative irreducible matrix with R, the largest diagonal entry and r, the smallest diagonal entry. Let Y = (y1 , y2 , . . . , yn )T be the p-norm normalized principle eigenvector of B corresponding to the spectral radius ρ and y1 ≥ y2 ≥ · · · ≥ yn . If p ≥ 2, then
y
maxp
(n − 1)(p−2)/2 (ρ − r)p/2 = y1 ≤ (n − 1)(p−2)/2 (ρ − r)p/2 + (ρ − R)p/2
1/p (13)
with equality holding in (13) if and only if B = c Ω, where c > 0 and the matrix Ω is defined as Ω = [ωi, j ], where
ωi, j
⎧ ⎪ R/c ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r/c ⎪ ⎪ ⎨ = 1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩0
if i = j = 1, if i = j ≥ 2, if i = 1, j ≥ 2, if j = 1, i ≥ 2, if i ≥ 2, j ≥ 2, i = j.
We now give an upper bound for the largest entry of the principal eigenvector of the distance signless Laplacian matrix of G. Corollary 3.2. Let G be a connected graph and also let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector associated to the spectral radius ∂ Q of DQ (G) with y1 ≥ y2 ≥ · · · ≥ yn . If p ≥ 2, then
K.Ch. Das et al. / Linear Algebra and its Applications 483 (2015) 200–220
y maxp = y1 ≤
(n − 1)(p−2)/2 (∂ Q − t)p/2 (p−2)/2 (n − 1) (∂ Q − t)p/2 + (∂ Q − T )p/2
213
1/p ,
(14)
where T and t denote, respectively, the maximal and minimal transmission of G. The equality occurs if and only if G ∼ = K2 . Proof. Setting R = T , r = t and ρ = ∂ Q in (13), we can get the required result in (14). The equality occurs in (14) if and only if there is a graph G such that its distance signless Laplacian matrix is written as DQ (G) = c Ω, where c > 0 and Ω is defined in Lemma 3.1. Since all the entries of the distance signless Laplacian matrix are greater than or equal to 1, it follows that G ∼ = K2 as G is connected graph and ωi, j = 0, i, j ≥ 2, i = j. Since c = r = R = 1, we have DQ (K2 ) = c Ω, the result is proven. 2 Remark 3.3. If p = 2 and the graph is transmission regular, the bound becomes y maxp ≤ √ 2 2 . Analogously to the result stated in [6] for distance matrix, we present, in the sequence, upper and lower bounds for the maximal entry of the principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). The lower bound is mentioned in Theorem 1.5. The proof is as follows: Proof of Theorem 1.5. Since Y is the p-normalized principal eigenvector associated to the spectral radius ∂ Q of DQ (G) and y1 ≥ y2 ≥ · · · ≥ yn , we have n
1=
ykp ≤ y1p + ynp + (n − 2) y2p ,
k=1
that is, ynp ≥ 1 − y1p − (n − 2) y2p .
(15)
Moreover, DQ Y = ∂ Q Y. For v1 ∈ V (G), ∂ Q y1 = T1 y1 +
n
d1, k yk ≥ t y1 +
k=2
n
yk + (n − deg G (v1 ) − 1) yn .
(16)
k=2
Since p ≥ 1, we multiply both sides of (16) by y2p−1 , we get ∂
Q
y1 y2p−1
≥
t y1 y2p−1
+
n
yk y2p−1 + (n − deg G (v1 ) − 1) yn y2p−1
k=2
≥ t y1 y2p−1 +
n k=2
ykp + (n − deg G (v1 ) − 1) ynp
(17)
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as y2 ≥ y3 ≥ · · · ≥ yn
" # ≥ t y2p−1 y1 + 1 − y1p + (n − deg G (v1 ) − 1) 1 − y1p − (n − 2) y2p ≥ t y2p−1 y1 + (n − Δ) (1 − y1p ) − (n − 2) (n − Δ − 1) y2p as deg G (v1 ) ≤ Δ ≥ t y2p−1 y1 + (n − Δ) (1 − y1p ) − (n − 2) (n − Δ − 1) y2p−1 as p ≥ 1. (18) Thus we have
y2 ≥
(n − Δ) (1 − y1p ) (∂ Q − t) y1 + (n − 2) (n − Δ − 1)
1/(p−1) .
(19)
For v2 ∈ V (G), we have n
∂ Q y2 = T2 y2 +
d2, k yk ≤ T y2 + T y1 .
(20)
k=1, k=2
From (19) and (20), we get (n − Δ) (1 − y1p ) (∂ Q − T )p−1 ≤ T p−1 y1p−1 , (∂ Q − t) y1 + (n − 2) (n − Δ − 1) that is, (n − Δ) (∂ Q − T )p−1 − (n − 2) (n − Δ − 1) T p−1 y1p−1 $ % ≤ (∂ Q − t) T p−1 + (n − Δ)(∂ Q − T )p−1 y1p $ % ≤ (∂ Q − t) T p−1 + (n − Δ)(∂ Q − T )p−1 y1p−1 ,
(21)
which gives the required result in (3). Now we have to prove that the inequality is strict. Since G is connected, yi > 0, i = 1, 2, . . . , n. Moreover, 2t ≤ ∂ Q ≤ 2T . Thus we have Q ∂ − t > 0, T > 0 and y1p < y1p−1 . Therefore one can see easily that (21) is strict. This completes the proof. 2 Corollary 3.4. Let G be a connected graph. Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then y maxp ≥
∂Q p−1
(2T )
1/p
p−1
(∂ Q − t) + ∂ Q
p−1
,
(22)
where T and t denote the maximal and minimal transmissions of G, respectively. Moreover, the equality holds in (22) if and only if G ∼ = Kn .
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215
Proof. From (17), we get ∂ Q y1 y2p−1 ≥ 1 − y1p + t y2p−1 y1 ,
(23)
that is,
1 − y1p y1 (∂ Q − t)
y2 ≥
1/p−1 .
(24)
From (20), we get ∂ Q y2 ≤ 2 T y1 .
(25)
From the above two results, we get
∂
Q
1 − y1p y1 (∂ Q − t)
1/p−1 ≤ 2 T y1 ,
implying y1p ≥
∂Q p−1
(2T )
p−1
(∂ Q − t) + ∂ Q
p−1 ,
which gives the required result in (22). The first part of the proof is done. Now suppose that equality holds in (22). Then all inequalities in the above must be equalities. In particular, from equality in (23), we get deg G (v1 ) = n − 1, T1 = t = n − 1 and y2 = y3 = · · · = yn . Again from equality in (25), we get T2 = T and y1 = y3 = y4 = · · · = yn . Thus we have y1 = y2 = · · · = yn . For vi ∈ V (G), ∂ Q = 2Ti , i = 1, 2, . . . , n; that is T1 = T2 = · · · = Tn and hence T = t = n − 1. Therefore deg G (v1 ) = deg G (v2 ) = · · · = deg G (vn ) = n − 1. Hence G ∼ = Kn . Conversely, let G ∼ = Kn . Then we have ∂ Q = 2(n − 1), T = t = n − 1 and y1 = y2 = 1 1/p . Hence the equality holds in (22) for Kn . 2 · · · = yn = n Analogously to what have been done for y minp , we also present in the next Corollary, bounds for y maxp do not depending on its largest eigenvalue. Corollary 3.5. Let G be a connected graph. Fixed p ≥ 3, let Y = (y1 , y2 , . . . , yn )T the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then
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y
maxp
≥
1/p
p−1
(2 t) p−1
(2 T )
,
p−1
t + (2 t)
where T and t denote the maximal and minimal transmissions of G, respectively. If p = 1 or p = 2, then y
maxp
≥
1/p
p−1
(2 T ) p−1
(2 T )
p−1
(2 T − t) + (2 T )
.
In any case, the equality occurs if and only if G ∼ = Kn . Proof. Fixed a > 0, consider the function f (x) =
xp−1 , a (x − t) + xp−1
2 t ≤ x ≤ 2 T.
Then we have f (x) =
a xp−2 ((p − 1) (x − t) − x) (a (x − t) + xp−1 )2
(26)
and f (x) = 0 if and only if x =
t (p − 1) . p−2
It is easy to see that f (x) ≥ 0 as x ≥ 2t ≥
t (p − 1) for p ≥ 3. p−2
If p = 1 or p = 2, then from (26), we get f (x) ≤ 0. Using these results together with the fact that 2t ≤ ∂ Q ≤ 2T and Corollary 3.4 we obtain the lower bound. Moreover, in any case, the equality holds if and only if G ∼ = Kn , by Corollary 3.4. 2 Lemma 3.6. Let G be an r-regular graph of diameter 2. Fix p ≥ 1 and let Y = (y1 , y2 , . . . , yn )T be the p-normalized principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G) with y maxp = y1 ≥ y2 ≥ · · · ≥ yn . Then T1 = T2 = · · · = Tn = 2n − r − 2, ∂ Q (G) = 2(2n − r − 2)
1/p 1 . and y1 = y2 = · · · = yn = n
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217
Proof. Since G is r-regular graph with d = 2, we have Ti = r + 2(n − 1 − r) = 2(n − 1) − r and hence T1 = T2 = · · · = Tn = T , (say). For vi ∈ V (G), n
∂ Q yi = Ti yi +
n
di, k yk , that is, (∂ Q − T ) yi =
k=1, k=i
di, k yk .
k=1, k=i
Taking summation from i = 1 to n, we get (∂ − T ) Q
n
yi =
i=1
n
n
di, k yk
i=1 k=1, k=i
=
n
[r + (n − 1 − r)2] yi
i=1
= (2n − 2 − r)
n
yi .
i=1
Since n
yi > 0,
i=1
from the above, we get ∂ Q (G) = T + 2n − r − 2. The first part of the proof is done. For v1 ∈ V (G), ∂ Q y1 = T1 y1 +
n
d1, k yk ≤ T y1 + (2n − 2 − r) y2 .
(27)
k=2
Since ∂ Q (G) = T + 2n − r − 2, from the above, we get y1 ≤ y2 . But we have y1 ≥ y2 . Thus we have y1 = y2 and hence from (27), y1 = y2 = · · · = yn . Since n
yip = 1,
i=1
we have y1 = y2 = · · · = yn =
1 1/p n
.
2
In Theorem 1.6, we mentioned an upper bound for the maximal entry of the principal eigenvector corresponding to the spectral radius ∂ Q of DQ (G). The proof is as follows:
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218
Proof of Theorem 1.6. For vi ∈ V (G), n
∂ Q yi = Ti yi +
di, k yk ,
k=1, k=i
that is, (∂ Q − T )yi ≤ (∂ Q − Ti )yi =
n
di, k yk .
(28)
k=1, k=i
Since p ≥ 1, by weighted power mean inequality, we get ⎛
n
⎜ k=1, k=i ⎜ n ⎝
⎛
⎞1/p
di, k ykp
di, k yk
⎜ k=1, k=i ≥⎜ n ⎝
⎟ ⎟ ⎠
di, k
⎞
n
k=1, k=i
di, k
⎟ ⎟, ⎠
k=1, k=i
that is, ⎛
⎞p
n
⎝
⎛
di, k yk ⎠ ≤ ⎝
k=1, k=i
⎞p−1
n
di, k ⎠
k=1, k=i
n
di, k ykp .
(29)
k=1, k=i
Since n
ykp = 1,
k=1
from (28) and (29), we get (∂ − T ) Q
p
yip
≤
Tip−1
(1 −
yip )
+
n
(di, k −
1) ykp
.
(30)
k=1, k=i
Setting i = 1 in (30), we get (∂ − T ) Q
p
y1p
≤T
p−1
≤T
p−1
(1 −
y1p )
+ (d − 1) (n − deg G (v1 ) −
1) y2p
as T ≥ T1
(1 −
y1p )
+ (d − 1) (n − δ −
1) y2p
.
(31)
Similarly, setting i = 2 in (30), we get (∂ Q −
T )p y2p
≤ T p−1 (1 −
y2p )
+ (d − 1) (n − δ −
1) y1p
,
(32)
K.Ch. Das et al. / Linear Algebra and its Applications 483 (2015) 200–220
219
that is, y2p ≤
(d − 1) (n − δ − 1) y1p + 1 p−1 T . (∂ Q − T )p + T p−1
Putting this value in (31), we get
(∂ − T ) + T Q
p
p−1
+ (d − 1) (n − δ − 1) T
y1p ≤
(∂ Q
T p−1
2
(∂ Q − T )p + T p−1 =
p−1
−
T )p
+
T p−1
− (d − 1)2 (n − δ − 1)2 T 2p−2
T p−1 , − (d − 1) (n − δ − 1) T p−1
which gives the required result in (4). The first part of the proof is done. Now suppose that equality holds in (4). Then all inequalities in the above must be equalities. In particular, from equality in (28), we get Ti = T . From equality in (29), we get y1 = y2 = · · · = yi−1 = yi+1 = · · · = yn . From equality in (31), we get deg G (v1 ) = δ, T1 = T , y2 = y3 = · · · = yn and d ≤ 2. From equality in (32), we get deg G (v2 ) = δ, T2 = T , y1 = y3 = · · · = yn and d ≤ 2. Thus we have y1 = y2 = · · · = yn = y, (say). If d = 1, G ∼ = Kn . Otherwise, d = 2. For vi ∈ V (G), " # ∂ Q y = 2Ti y, that is, ∂ Q = 2 deg G (vi ) + 2(n − 1 − deg G (vi )) = 4(n − 1) − 2 deg G (vi ). From the above, we conclude that deg G (v1 ) = deg G (v2 ) = · · · = deg G (vn ). Hence G is a regular graph of diameter 2. Conversely, let G ∼ = Kn . Then we have ∂ Q = 2(n − 1), T = n − 1 = δ, d = 1 and 1/p y1 = y2 = · · · = yn = n1 . One can see easily that equality holds in (4). Let G be an r-regular graph of diameter 2. Then by Lemma 3.6, we get d = 2, 1/p T1 = T2 = · · · = Tn = 2n − r − 2 = T , (say), y1 = y2 = · · · = yn = n1 and ∂ Q (G) = 2T . Then T p−1 T p−1 = (∂ Q − T )p + T p−1 − (d − 1)(n − δ − 1) T p−1 T p + T p−1 − (n − r − 1) T p−1 1 T −n+r+2 1 = = y1p as T = 2n − r − 2. n
=
Hence equality holds in (4). 2
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