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On incidence energy of graphs
Linear Algebra and its Applications 446 (2014) 329–344
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Linear Algebra and its Applications www.elsevier.com/locate/laa
On incidence energy of graphs Kinkar Ch. Das a , Ivan Gutman b,∗,1 a Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea b Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia
a r t i c l e
i n f o
Article history: Received 16 November 2013 Accepted 18 December 2013 Available online 7 January 2014 Submitted by R. Brualdi MSC: 05C50 15A18 Keywords: Graph spectrum Laplacian spectrum (of graph) Incidence matrix Incidence energy Energy (of matrix)
a b s t r a c t Let G = (V, E) be a simple graph with vertex set V = {v1 , v2 , . . . , vn } and edge set E = {e1 , e2 , . . . , em }. The incidence matrix I(G) of G is the n × m matrix whose (i, j)-entry is 1 if vi is incident to ej and 0 otherwise. The incidence energy IE of G is the sum of the singular values of I(G). In this paper we give lower and upper bounds for IE in terms of n, m, maximum degree, clique number, independence number, and the first Zagreb index. Moreover, we obtain Nordhaus–Gaddum-type results for IE. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Let G = (V (G), E(G)) be a simple graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G) = {e1 , e2 , . . . , em }. Let dG (vi ) be the degree of the vertex vi . The maximum and minimum degrees are denoted by Δ and δ, respectively. * Corresponding author. 1
E-mail addresses: [email protected] (K.Ch. Das), [email protected] (I. Gutman). Fax: +381 34 335040.
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.12.026
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For a subset V of V (G), the subgraph of G obtained by deleting the vertices of V and the edges incident with them is denoted by G−V . Similarly, G−E is the subgraph of G, obtained by deleting the edges from a subset E ⊂ E(G). If V = {vi } and E = {vj vk }, then G − V and G − E will be written as G − vi and G − vj vk , respectively. If vi and vj are nonadjacent vertices of the graph G, then G + vi vj denotes the graph obtained by adding to G a new edge vi vj . As usual, Kn and K1,n−1 stand, respectively, for the complete graph and the star on n vertices. Let A(G) be the (0, 1)-adjacency matrix of G and D(G) be the diagonal matrix of vertex degrees. The Laplacian matrix of G is L(G) = D(G) − A(G). This matrix has nonnegative eigenvalues n μ1 μ2 · · · μn = 0. Denote by Spec(G) = {μ1 , μ2 , . . . , μn } the spectrum of L(G), i.e., the Laplacian spectrum of G. When more than one graph is under consideration, then we write μi (G) instead of μi . The signless Laplacian matrix of G is Q(G) = D(G)+A(G). This matrix has nonnegative eigenvalues q1 q2 · · · qn 0. For a connected graph G, qn = 0 if and only if G is bipartite. In this case, 0 is a simple eigenvalue. As well known [34], n i=1
qi =
n
μi = 2m.
(1)
i=1
The incidence matrix of G is the n × m matrix I(G) whose (i, j)-entry is 1 if vi is incident to ej and 0 otherwise. It is known that [7] I(G)I(G)t = D(G) + A(G) = Q(G).
(2)
The notion of the energy of a graph G was introduced by one of the present authors in 1978 as the sum of the absolute values of the eigenvalues of A(G). It’s origin was from chemistry, where it is connected with the total π-electron energy of a molecule [24]. Eventually, it was recognized that the interest for this graph invariant goes far beyond chemistry; see the reviews [18,23], the recent monograph [30], and the recent papers [15,39]. For lower and upper bounds on graph energy see [12,13,20,26,30] and the references cited therein. In view of the success of the mathematical theory of graph energy, several energy-like invariants were put forward, based on graph matrices different than A(G). In [32], the so-called Laplacian-energy-like invariant, LEL, was defined as LEL = LEL(G) =
n √
μi .
(3)
i=1
For its basic properties see the survey [31]. Upper and lower bounds on LEL were reported in [4,10,11,14,32,42]. Another energy-like quantity in the incidence energy, IE, invented by Jooyandeh et al. [29]. This graph invariant is defined as the sum of the singular values σi of the
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incidence matrix I(G). Recall that the singular values of a matrix M are the nonnegative square roots of MMt or Mt M and that these matrices have the same nonzero eigenvalues. From these facts, and bearing in mind identity (2), it follows that [21,22] IE = IE(G) =
n √
qi .
(4)
i=1
The mathematical theory of incidence energy is nowadays elaborated in considerable detail, see [5,6,21,22,28,33,35,38,41,43,44]. The present paper is organized as follows. In Section 2, we give a list of some previously known results. In Section 3, we present novel lower bounds for IE in terms of n, m, Δ, and clique number. In Section 4, we give novel upper bounds for IE in terms of n, m, independence number, and the first Zagreb index. In Section 5, we deduce some Nordhaus–Gaddum-type results for IE. 2. Preliminaries We state here some previously known results that will be needed in the subsequent three sections. Lemma 2.1 (Jensen’s Inequality). (See [25].) If f is convex on an interval I and x1 , x2 , . . . , xn are in I, then
x1 + x2 + · · · + xn f n
f (x1 ) + f (x2 ) + · · · + f (xn ) . n
For strictly convex functions, equality holds if and only if x1 = x2 = · · · = xn . Lemma 2.2. (See [40].) Let B be a p × p symmetric matrix and let Bk be its leading k × k submatrix. Then, for i = 1, 2, . . . , k, λp−i+1 (B) λk−i+1 (Bk ) λk−i+1 (B)
(5)
where λi (B) is the i-th greatest eigenvalue of B. Lemma 2.3. (See [27].) The Laplacian eigenvalues of G interlace the Laplacian eigenvalues of G + e, i.e., μ1 (G + e) μ1 (G) μ2 (G + e) · · · μn (G + e) μn (G). Lemma 2.4. (See [34].) Let G be a graph with Laplacian spectrum {0 = μn , μn−1 , . . . , μ1 }. Then the Laplacian spectrum of G is {0, n − μ1 , n − μ2 , . . . , n − μn−1 }, where G is the complement of G.
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Lemma 2.5. (See [1,2].) For any α (0 < α 1 or 2 α < 3), α α q1α + q2α + · · · + qnα μα 1 + μ2 + · · · + μn .
Moreover, the equality holds for α ∈ / {1, 2} if and only if G is isomorphic to a bipartite graph. Lemma 2.6. (See [17].) Let G be a simple graph on n vertices with at least one edge. Then q1 Δ + 1. If G is connected, then the equality holds if and only if G ∼ = K1,n−1 . Lemma 2.7. (See [36].) Let G be a simple graph on n vertices with m edges. Then q1 4m/n with equality if and only if G is regular. Lemma 2.8. (See [8].) Let G be a connected graph of order n. Then qn < δ. We denote the set of neighbors of the vertex vi by Ni = {vj ∈ V (G) | vi vj ∈ E(G)}. If G has distinct vertices vi and vj with Ni = Nj , then vi and vj are said to be duplicates and (vi , vj ) is called a duplicate pair [16]. Lemma 2.9. (See [9].) Let G = (V, E) be a graph with vertex set V (G) = {v1 , v2 , . . . , vn }. If G has k − 1 duplicate pairs (vi , vi+1 ), i = 1, 2, . . . , k − 1, then this graph G has at least k − 1 equal signless Laplacian eigenvalues and they are all equal to the cardinality of the neighbor set. The corresponding k − 1 eigenvectors are of the form ( 1, −1, 0, . . . , 0)t , ( 1, 0, −1, 0, . . . , 0)t , . . . , ( 1, 0, . . . , −1, 0, . . . , 0)t . 2
3
k
Lemma 2.10. Let G be a graph with vertex set {v1 , v2 , . . . , vn }. Let G have a clique of size ω whose vertices are v1 , v2 , . . . , vω . If dG (vi ) = ω − 1 for i = 2, 3, . . . , ω, then G has at least ω − 2 mutually equal signless Laplacian eigenvalues, equal to ω − 2. The corresponding ω − 2 eigenvectors are (0, 1, −1, 0, . . . , 0)t , (0, 1, 0, −1, 0, . . . , 0)t , . . . , (0, 1, 0, . . . , −1, 0, . . . , 0)t . 2
3
ω−1
Proof. Let x = (x1 , x2 , . . . , xn )t be an eigenvector of Q(G), corresponding to an eigenvalue q(G). Then for i = 1, 2, . . . , n, q(G)xi = dG (vi )xi +
xj .
j: vi vj ∈E(G)
One can see easily that the eigenvalue ω − 2 with corresponding eigenvectors (0, 1, −1, 0, . . . , 0)t , (0, 1, 0, −1, 0, . . . , 0)t , . . . , (0, 1, 0, . . . , −1, 0, . . . , 0)t 2
3
ω−1
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satisfies the above relation. Since these eigenvectors are linearly independent, we conclude that ω − 2 is an eigenvalue of Q(G) with multiplicity at least ω − 2. 2 Lemma 2.11. (See [38].) Let e ∈ E(G). Then, IE(G − e) < IE(G). 3. Lower bounds on incidence energy In this section, we give two lower bounds on IE in terms of n, m, Δ, clique number ω, and the number t of spanning trees. Denote by KS n,ω the graph of order n with clique number ω, obtained by attaching n − ω pendent edges to one vertex of the complete graph Kω . In order to deduce our first lower bound, we need the following results: Lemma 3.1. Let KS n,ω be the graph defined above. Then its signless Laplacian eigenvalues are a, b, ω − 2, ω − 2, . . . , ω − 2, 1, 1, . . . , 1, c ω−2
n−ω−1
where a, b, and c (a b c) are the roots of the equation: x3 − (2ω + n − 3)x2 + n(2ω − 3)x − (ω − 1)(2ω − 4) = 0. Proof. By Lemmas 2.9 and 2.10, we conclude that ω − 2, ω − 2, . . . , ω − 2, 1, 1, . . . , 1 ω−2
n−ω−1
are signless Laplacian eigenvalues of KS n,ω . The remaining eigenvalues satisfy: qx1 = (n − 1)x1 + (ω − 1)x2 + (n − ω)x3 , qx2 = (ω − 1)x2 + (ω − 2)x2 + x1 , qx3 = x3 + x1 . Thus the remaining signless Laplacian eigenvalues of KS n,ω satisfy Eq. (6). 2 Corollary 3.2. Let KS n,ω be the graph defined above. Then IE(KS n,ω ) =
√
a+
√
b+
√
c + (ω − 2)3/2 + n − ω − 1
where a, b, and c are the same as in Lemma 3.1. We are now ready to give a lower bound on IE(G) in terms of n, Δ, and ω.
(6)
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Theorem 3.3. Let G ( Kn ) be a connected graph of order n with maximum degree Δ and clique number ω. Then IE(G)
√
a+
√
b+
√
c + (ω − 2)3/2 + Δ − ω,
(7)
where a, b and c (a b c) are the roots of the equation g(x) = 0, where g(x) = x3 − (2ω + Δ − 2)x2 + (Δ + 1)(2ω − 3)x − (ω − 1)(2ω − 4). Moreover, the equality holds in (7) if and only if G ∼ = KS n,ω . Proof. Since G Kn , we have ω n − 1. We now consider two cases (i) Δ = n − 1, (ii) Δ n − 2. ∼ KS n,ω , then by Case (i): Δ = n − 1. In this case KS n,ω is a subgraph of G. If G = Corollary 3.2, equality holds in (7). Otherwise, |E(G)| > |E(KS n,ω )| and by Lemma 2.11 and Corollary 3.2, IE(G) > IE(KS n,ω ) =
√
a+
√
b+
√
c + (ω − 2)3/2 + Δ − ω.
Case (ii): Δ n − 2. Since ω n − 2, we have Δ ω as G is connected. First we assume that the maximum degree vertex vi is in V (Kω ) of graph G. In this case KS Δ+1,ω is a subgraph of G and |E(G)| > |E(KS Δ+1,ω )| as G is connected. By Lemma 2.11 and Corollary 3.2, we get IE(G) > IE(KS Δ+1,ω ) =
√
a+
√
b+
√
c + (ω − 2)3/2 + Δ − ω.
Assume next that the maximum degree vertex vi is in V (G) \ V (Kω ). Let S be the set of vertices in Kω of G. Then S = V (Kω ) = {v1 , v2 , . . . , vω }. Suppose that NG (vi ) ∩ S = {v1 , v2 , . . . , vp }. Then |NG (vi ) ∩ S| = p. Since ω is the clique number of G, we must have 0 p ω − 1. (For p = ω, the clique number would be ω + 1, a contradiction.) If p = ω − 1, then the ω − 1 = p vertices in S, together with the maximum degree vertex vi , form a clique Kω . The maximum degree vertex vi is in this clique. Hence we get the earlier result (7). Otherwise, 0 p ω − 2. Then Kω ∪ K1,Δ−p is a subgraph of G. The signless Laplacian spectrum of Kω ∪ K1,Δ−p is
2(ω − 1), ω − 2, ω − 2, . . . , ω − 2, Δ − p + 1, 1, 1, . . . , 1, 0, 0 . ω−1
Δ−p−1
Let g(x) be the function defined in Theorem 3.3. Since a, b, and c are the roots g(x) = 0, we have
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a + b + c = 2ω + Δ − 2,
335
(8)
ab + bc + ca = (Δ + 1)(2ω − 3),
(9)
abc = (ω − 1)(2ω − 4). Now, g(0) = −(ω −1)(2ω −4) < 0 and g(1) = 2(ω −2)(Δ−ω +1) > 0. Since a b c, we must have 0 < c < 1. We have to show that √
a+
√
b<
2(ω − 1) +
√
Δ−ω+3+
√
ω−2
(10)
that is, √ a + b + 2 ab < 2ω + Δ − 1 + 2 2(ω − 1)(Δ − ω + 3) + 2(ω − 1)(ω − 2)
+ (ω − 2)(Δ − ω + 3) that is, ab <
1 + 2(ω − 1)(Δ − ω + 3) + 2(ω − 1)(ω − 2) + (ω − 2)(Δ − ω + 3) 4
by (8)
that is, ab < 2(ω − 1)(Δ + 1) which by (9) is always true. Since 0 p ω − 2, we have Δ − p − 1 Δ − ω + 1. Now, IE(G) IE(Kω ∪ K1,Δ−p ) √ = Δ − p + 1 + 2(ω − 1) + (ω − 1) ω − 2 + Δ − p − 1 √ √ Δ − ω + 3 + 2(ω − 1) + (ω − 1) ω − 2 + Δ − ω + 1 √ √ √ √ > a + b + c + (ω − 2) ω − 2 + Δ − ω as c < 1 and by (10). This completes the proof. 2 The following result was obtained in [10]: Lemma 3.4. Let G be a connected graph of order n with m edges, maximum degree Δ, and t spanning trees. Then LEL(G)
√
Δ + 1 + 2m − n + (n − 2)(n − 3)t1/(n−2)
with equality if and only if G ∼ = Kn or G ∼ = K1,n−1 .
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We now establish the incidence-energy analogue of Lemma 3.4: Theorem 3.5. Let G be a connected graph of order n with m edges, maximum degree Δ, and t spanning trees. Then √ IE(G) Δ + 1 + 2m − n + (n − 2)(n − 3)t1/(n−2) (11) with equality holding in (11) if and only if G ∼ = K1,n−1 . Proof. Bearling in mind Eqs. (3) and (4), Lemma 2.5 implies IE(G) =
√
q1 +
√
q2 + · · · +
√
qn
√
μ1 +
√
μ2 + · · · +
√ μn = LEL(G)
(12)
which together with Lemma 3.4 yields (11). Moreover, by Lemma 2.5, equality holds in (12) if and only if G is isomorphic to a bipartite graph. Hence equality holds in (11) if and only if G ∼ = K1,n−1 . 2 4. Upper bounds on incidence energy In this section, we give two upper bounds on IE in terms of n, m, independence number α, and first Zagreb index M1 . The first Zagreb index M1 = M1 (G) is equal to the sum of squares of the vertex degrees of the graph G [19]. Theorem 4.1. Let G be a connected graph of order n with m edges. Then 1/4 16m2 4m 3/4 M1 (G) + 2m − + (n − 1) IE(G) . n n2
(13)
Equality holds if and only if G ∼ = Kn . Proof. We start with n
qi2 =
i=1
n
dG (vi ) dG (vi ) + 1 = M1 + 2m.
i=1
Assume first that G is non-bipartite. Since G is connected, we have qi > 0, i = 1, 2, . . . , n. Then, by applying Lemma 2.1,
1 √ q n − 1 i=2 n
4
1 2 M1 + 2m − q12 qi = n − 1 i=2 n−1 n
that is n √ i=2
1/4 qi (n − 1)3/4 M1 + 2m − q12
(14)
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with equality holding if and only if q2 = q3 = · · · = qn . This implies
IE(G) =
√
q1 +
n √
qi
i=2
√
1/4 q1 + (n − 1)3/4 M1 (G) + 2m − q12 .
(15)
Consider now the case when G is bipartite. Since G is connected, qn = 0 and qi > 0, i = 1, 2, . . . , n − 1. This time, instead of (14) we have n √
1/4 qi (n − 2)3/4 M1 (G) + 2m − q12
i=2
1/4 < (n − 1)3/4 M1 (G) + 2m − q12 and we again arrive at the inequality (15). Bearing in mind the bound (15), define an auxiliary function f (x) =
1/4 √ x + (n − 1)3/4 M1 (G) + 2m − x2
whose first derivative is 1 (n − 1)3/4 x f (x) = √ − . 2 x 2(M1 (G) + 2m − x2 )3/4 Since nq12
n
qi2 = M1 (G) + 2m
i=1
one can easily verify that f (x) 0 i.e., that f (x) is a decreasing function. Then by using Lemma 2.7, we get
4m f (x) f n
=
1/4 16m2 4m + (n − 1)3/4 M1 (G) + 2m − n n2
which combined with (15) yields (13). First part of the proof is done. Suppose now that equality holds in (13). Then all inequalities in the above argument must be equalities. This requires that q2 = q3 = · · · = qn . Moreover, it also must be q1 = 4m/n and thus by Lemma 2.7 G must be regular.
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By contradiction, we show that G is isomorphic to Kn . For this suppose that G is a Δ-regular graph, Δ = n − 1. Then there exist two vertices vi and vj in G that are not adjacent. By Lemma 2.2, q1 q1 and q2 q2 , where q1 and q2 are the solutions of Δ−q 0 = 0. 0 Δ−q Therefore, q1 Δ and q2 Δ. By Lemma 2.8, qn < Δ, as G is a regular graph different from Kn . Then, however, Δ q2 = qn < Δ, which is a contradiction. Therefore, it must be G ∼ = Kn . Conversely, let G ∼ = Kn . Then
1/4 √ 16m2 4m 3/4 M1 (G) + 2m − + (n − 1) = 2(n − 1) + (n − 1) n − 2 = IE(G). n n2
This completes the proof. 2 Corollary 4.2. Let G be a connected graph of order n with m edges and maximum degree Δ. Then IE(G) <
√
1/4 Δ + 1 + (n − 1)3/4 M1 (G) + 2m − (Δ + 1)2 .
(16)
Proof. The function f (x) in Theorem 4.1 is decreasing. By Lemma 2.6, q1 Δ + 1. Using these results, we get (16). Suppose now that the equality holds in (16). Moreover, from Theorem 4.1 we know that q1 = Δ + 1 and q2 = q3 = · · · = qn . Since G is connected, by Lemma 2.6, we have G ∼ = K1,n−1 and hence 1 = q2 = qn = 0, a contradiction. This completes the proof. 2 A complete split graph CS(n, ω), ω n, is a graph on n vertices consisting of a clique on ω vertices and a stable set on the remaining n − ω vertices, such that each vertex of the clique is adjacent to each vertex of the stable set. Given a graph G, a subset S of V (G) is called an independent set of G if G[S], the subgraph induced by S, is a graph with |S| isolated vertices. The independence number of G is denoted by α and is defined to be the number of vertices in the largest independent set of G. The signless Laplacian eigenvalues of the complete split graph CS(n, n − α), 1 α n − 1, are [9]:
1 3n − 2α − 2 ± (n − 2)2 + 4α(n − α) , n − 2, . . . , n − 2, n − α, . . . , n − α, 2 n−α−1
α−1
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from which it straightforwardly follows √ √ IE CS(n, n − α) = (n − α − 1) n − 2 + (α − 1) n − α 1 +√ 3n − 2α − 2 + (n − 2)2 + 4α(n − α) 2 1 +√ 3n − 2α − 2 − (n − 2)2 + 4α(n − α). 2 Theorem 4.3. Let G be a graph of order n with independence number α. Then IE(G) IE CS(n, n − α) with equality if and only if G ∼ = CS(n, n − α). Proof. By Lemma 2.11, if a new edge is inserted to the graph G, then its independence energy will increase. Adding to G edges one-by-one, so that its α-value is not changed, we arrive at CS(n, n − α). Thus, if G CS(n, n − α), then IE(G) < IE(CS(n, n − α)). 2 5. Nordhaus–Gaddum-type results for incidence energy Nordhaus and Gaddum [37] gave bounds for the sum of the chromatic numbers of a graph and its complement. Eventually, numerous Nordhaus–Gaddum-type results for other graph invariants were obtained [3]. With regard to incidence energy, the following result is known [22]: Lemma 5.1. Let G be a graph of order n 2. Then √ √ (n − 1) n IE(G) + IE(G) < 2 n − 1 + (n − 1) 2(n − 2) with left equality if and only if n = 2. Here we give a few more Nordhaus–Gaddum-type results for IE. First we give a lower bound on IE(G) + IE(G) in terms of n and the number of spanning trees t and t for G and G, respectively. Theorem 5.2. Let G be a graph of order n 2. Then 1/[2(n−1)] IE(G) + IE(G) (n − 1) nn−1 + n2n−1 tt .
(17)
Equality holds if and only if G ∼ = K2 . Proof. For n = 2, one can see easily that the inequality holds in (17) and that equality holds in (17) for G ∼ = K2 . Let, therefore, n 3. We have μi 0 and n − μi 0, that is μi (n − μi ) 0, i = 1, 2, . . . , n − 1.
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One can see easily that
n + 2 μ1 (n − μ1 ) n + 2 μ2 (n − μ2 ) · · · n + 2 μn−1 (n − μn−1 ) n−1 n−1 n−1 n−1 n +2 μi × (n − μi ). i=1
i=1
Since t = t(G) =
n−1 1 μi n i=1
and t = t(G) =
n−1 1 (n − μi ) n i=1
from the above, we get n−1
n + 2 μi (n − μi ) nn−1 + 2n−1 n2 tt.
(18)
i=1
By the arithmetic–geometric mean (AGM) inequality,
n−1 1 n + 2 μi (n − μi ) n − 1 i=1
n−1
n−1
n + 2 μi (n − μi )
1/2 .
(19)
i=1
Now, IE(G) + IE(G) =
n √ ( qi + qi ) i=1
n √ ( μi + μi )
by Lemma 2.5
(20)
i=1
=
n−1
√ √ ( μi + n − μi ) by Lemma 2.4
i=1
=
n−1
n + 2 μi (n − μi )
i=1
n−1 1/[2(n−1)] (n − 1) n + 2 μi (n − μi )
by (19).
i=1
Using (18), we arrive at inequality (17). Suppose now that equality holds in (17). Then all inequalities in the above argument must be equalities. From the equality in (20), by Lemma 2.5, we get that both G and G are bipartite. From the equality in (19), it follows μ1 (n − μ1 ) = μ2 (n − μ2 ) = · · · = μn−1 (n − μn−1 )
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that is, μi (n − μi ) = μj (n − μj ) that is, (μi − μj )(μi + μj − n) = 0 for 1 i < j n. This implies that either μi = μj or μi + μj = n. Therefore, G must be isomorphic to a graph H such that μ1 (H) = μ2 (H) = · · · = μp (H) and μp+1 (H) = μp+2 (H) = · · · = μn−1 (H) = n − μ1 (H). Consequently, G is bipartite. From the equality in (18), we get
p n−p−1 n + 2 μ1 (H) n − μ1 (H) × n + 2 μn−1 (H) n − μn−1 (H) p n−p−1 n−p−1 = nn−1 + 2n−1 μp1 (H)μn−1 (H) n − μ1 (H) n − μn−1 (H)
that is, n−1 n + 2 μ1 (H)μn−1 (H) = nn−1 + 2n−1 μn−1 (H)μn−1 1 n−1 (H). Since n 3, one can see easily that n−1 n + 2 μ1 (H)μn−1 (H) > nn−1 + 2n−1 μn−1 (H)μn−1 1 n−1 (H). This completes the proof. 2 Remark 5.3. The lower bound (17) is always better than the lower bound in Lemma 5.1. We now give an upper bound on IE(G) + IE(G) in terms of n, m, Δ, and δ. Theorem 5.4. Let G be a graph of order n 2, with m edges, maximum degree Δ and minimum degree δ. Then IE(G) + IE(G)
√
2Δ + 2(n − δ − 1)
(n − 1)(n − 2) √ 2m + n(n − 1) − 2m + n
with equality if and only if G ∼ = Kn .
(21)
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Proof. By the Cauchy–Schwarz inequality,
IE(G) + IE(G) =
√
q1 +
q1 +
n √
qi +
i=2
√
2Δ +
n
qi
i=2
n n 2(n − δ − 1) + (n − 1) qi + (n − 1) qi i=2
(22)
i=2
as q1 2Δ and q 1 2(n − δ − 1) √ = 2Δ + 2(n − δ − 1) + (n − 1)(2m − q1 ) + (n − 1)(2m − q 1 ) as
n
qi = 2m and
i=1
n
q i = 2m
i=1
√ 2Δ + 2(n − δ − 1) +
4m 4m + (n − 1) 2m − (n − 1) 2m − n n
(23)
4m 4m and q 1 , by Lemma 2.7 n n √
(n − 1)(n − 2) √ = 2Δ + 2(n − δ − 1) + 2m + n(n − 1) − 2m n as q1
as 2m + 2m = n(n − 1). The first part of the proof is done. Suppose now that equality holds in (21). Then all inequalities in the above argument must be equalities. From the equality in (22), we get q1 = 2Δ, q1 = 2(n − δ − 1), q2 = q3 = · · · = qn , and q 2 = q 3 = · · · = q n . From the equality in (23), we get q1 = 4m/n and q1 = 4m/n. By Lemma 2.7, this implies that G and G are isomorphic to a regular graph. Assume first that G is connected. In a similar manner as in the proof of Theorem 4.1, we get G ∼ = Kn . If G is disconnected, then G is connected. Moreover, we have q 2 = q 3 = · · · = q n . As before, we can easily show that G ∼ = Kn . Hence G ∼ = K n. Conversely, one can see easily that the equality holds in (21) for complete graph Kn or for null graph K n . 2 Remark 5.5. For the complete graph Kn , the upper bound (21) is better than the upper bound in Lemma 5.1.
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Acknowledgements The first author is supported by the Faculty research Fund, Sungkyunkwan University, 2012, and National Research Foundation funded by the Korean Government with the grant No. 2013R1A1A2009341. References [1] S. Akbari, E. Ghorbani, J.H. Koolen, M.R. Oboudi, A relation between the Laplacian and signless Laplacian eigenvalues of a graph, J. Algebraic Combin. 32 (2010) 459–464. [2] S. Akbari, E. Ghorbani, J.H. Koolen, M.R. Oboudi, On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs, Electron. J. Combin. 17 (2010) R115. [3] M. Aouchiche, P. Hansen, A survey of Nordhaus–Gaddum type relations, Discrete Appl. Math. 161 (2013) 466–546. [4] B. Arsić, I. Gutman, K.C. Das, K. Xu, Relation between Kirchhoff index and Laplacian-energy-like invariant, Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur.) 144 (2012) 61–72. [5] Ş.B. Bozkurt, I. Gutman, Estimating the incidence energy, MATCH Commun. Math. Comput. Chem. 70 (2013) 143–156. [6] Ş.B. Bozkurt, D. Bozkurt, On incidence energy, MATCH Commun. Math. Comput. Chem. 72 (2014) 123–132. [7] D. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge, 2010. [8] K.C. Das, On conjectures involving second largest signless Laplacian eigenvalue of graphs, Linear Algebra Appl. 432 (2010) 3018–3029. [9] K.C. Das, Proofs of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs, Discrete Math. 312 (2012) 992–998. [10] K.C. Das, I. Gutman, A.S. Çevik, On the Laplacian-energy-like invariant, Linear Algebra Appl. 442 (2014) 58–68. [11] K.C. Das, I. Gutman, A.S. Çevik, B. Zhou, On Laplacian energy, MATCH Commun. Math. Comput. Chem. 70 (2013) 689–696. [12] K.C. Das, S.A. Mojallal, Upper bounds for the energy of graphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 657–662. [13] K.C. Das, S.A. Mojallal, I. Gutman, Improving McClelland’s lower bound for energy, MATCH Commun. Math. Comput. Chem. 70 (2013) 663–668. [14] K.C. Das, K. Xu, I. Gutman, Comparison between Kirchhoff index and Laplacian-energy-like invariant of graphs, Linear Algebra Appl. 436 (2012) 3661–3671. [15] W. Du, X. Li, Y. Li, The energy of random graphs, Linear Algebra Appl. 435 (2011) 2334–2346. [16] M.N. Ellingham, Basic subgraphs and graph spectra, Australas. J. Combin. 8 (1993) 247–265. [17] Y.Z. Fan, B.S. Tam, J. Zhou, Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order, Linear Multilinear Algebra 56 (2008) 381–397. [18] I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer, Berlin, 2001, pp. 196–211. [19] I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92. [20] I. Gutman, K.C. Das, Estimating the total π-electron energy, J. Serb. Chem. Soc. 78 (2013). [21] I. Gutman, D. Kiani, M. Mirzakhah, On incidence energy of graphs, MATCH Commun. Math. Comput. Chem. 62 (2009) 573–580. [22] I. Gutman, D. Kiani, M. Mirzakhah, B. Zhou, On incidence energy of a graph, Linear Algebra Appl. 431 (2009) 1223–1233. [23] I. Gutman, X. Li, J. Zhang, Graph energy, in: M. Dehmer, F. Emmert–Streib (Eds.), Analysis of Complex Networks. From Biology to Linguistics, Wiley–VCH, Weinheim, 2009, pp. 145–174. [24] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986. [25] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1988, pp. 83–84.
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[26] W.H. Haemers, Strongly regular graphs with maximal energy, Linear Algebra Appl. 429 (2008) 2719–2723. [27] J.V.D. Heuvel, Hamilton cycles and eigenvalues of graphs, Linear Algebra Appl. 226–228 (1995) 723–730. [28] A. Ilić, Trees with minimal Laplacian coefficients, Comput. Math. Appl. 59 (2010) 2776–2783. [29] M.R. Jooyandeh, D. Kiani, M. Mirzakhah, Incidence energy of a graph, MATCH Commun. Math. Comput. Chem. 62 (2009) 561–572. [30] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. [31] B. Liu, Y. Huang, Z. You, A survey on the Laplacian-energy-like invariant, MATCH Commun. Math. Comput. Chem. 66 (2011) 713–730. [32] J. Liu, B. Liu, A Laplacian-energy like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008) 397–419. [33] M. Liu, B. Liu, On sum of powers of the signless Laplacian eigenvalues of graphs, Hacet. J. Math. Stat. 41 (2012) 527–536. [34] R. Merris, Laplacian matrices of graphs: A survey, Linear Algebra Appl. 197 (198) (1994) 143–176. [35] M. Mirzakhah, D. Kiani, Some results on signless Laplacian coefficients of graphs, Linear Algebra Appl. 437 (2012) 2243–2251. [36] W. Ning, H. Li, M. Lu, On the signless Laplacian spectral radius of irregular graphs, Linear Algebra Appl. 438 (2013) 2280–2288. [37] E.A. Nordhaus, J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175–177. [38] O. Rojo, E. Lenes, A sharp upper bound on the incidence energy of graphs in terms of connectivity, Linear Algebra Appl. 438 (2013) 1485–1493. [39] J.W. Sander, T. Sander, Integral circulant graphs of prime order with maximal energy, Linear Algebra Appl. 435 (2011) 3212–3232. [40] J.R. Schott, Matrix Analysis for Statistics, Wiley, New York, 1997. [41] Z. Tang, Y. Hou, On incidence energy of trees, MATCH Commun. Math. Comput. Chem. 66 (2011) 977–984. [42] K. Xu, K.C. Das, Extremal Laplacian-energy-like invariant of graphs with given matching number, Electron. J. Linear Algebra 26 (2013) 131–140. [43] J. Zhang, J. Li, New results on the incidence energy of graphs, MATCH Commun. Math. Comput. Chem. 68 (2012) 777–803. [44] B. Zhou, More upper bounds for the incidence energy, MATCH Commun. Math. Comput. Chem. 64 (2010) 123–128.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
On incidence energy of graphs Kinkar Ch. Das a , Ivan Gutman b,∗,1 a Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea b Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia
a r t i c l e
i n f o
Article history: Received 16 November 2013 Accepted 18 December 2013 Available online 7 January 2014 Submitted by R. Brualdi MSC: 05C50 15A18 Keywords: Graph spectrum Laplacian spectrum (of graph) Incidence matrix Incidence energy Energy (of matrix)
a b s t r a c t Let G = (V, E) be a simple graph with vertex set V = {v1 , v2 , . . . , vn } and edge set E = {e1 , e2 , . . . , em }. The incidence matrix I(G) of G is the n × m matrix whose (i, j)-entry is 1 if vi is incident to ej and 0 otherwise. The incidence energy IE of G is the sum of the singular values of I(G). In this paper we give lower and upper bounds for IE in terms of n, m, maximum degree, clique number, independence number, and the first Zagreb index. Moreover, we obtain Nordhaus–Gaddum-type results for IE. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Let G = (V (G), E(G)) be a simple graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G) = {e1 , e2 , . . . , em }. Let dG (vi ) be the degree of the vertex vi . The maximum and minimum degrees are denoted by Δ and δ, respectively. * Corresponding author. 1
E-mail addresses: [email protected] (K.Ch. Das), [email protected] (I. Gutman). Fax: +381 34 335040.
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.12.026
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For a subset V of V (G), the subgraph of G obtained by deleting the vertices of V and the edges incident with them is denoted by G−V . Similarly, G−E is the subgraph of G, obtained by deleting the edges from a subset E ⊂ E(G). If V = {vi } and E = {vj vk }, then G − V and G − E will be written as G − vi and G − vj vk , respectively. If vi and vj are nonadjacent vertices of the graph G, then G + vi vj denotes the graph obtained by adding to G a new edge vi vj . As usual, Kn and K1,n−1 stand, respectively, for the complete graph and the star on n vertices. Let A(G) be the (0, 1)-adjacency matrix of G and D(G) be the diagonal matrix of vertex degrees. The Laplacian matrix of G is L(G) = D(G) − A(G). This matrix has nonnegative eigenvalues n μ1 μ2 · · · μn = 0. Denote by Spec(G) = {μ1 , μ2 , . . . , μn } the spectrum of L(G), i.e., the Laplacian spectrum of G. When more than one graph is under consideration, then we write μi (G) instead of μi . The signless Laplacian matrix of G is Q(G) = D(G)+A(G). This matrix has nonnegative eigenvalues q1 q2 · · · qn 0. For a connected graph G, qn = 0 if and only if G is bipartite. In this case, 0 is a simple eigenvalue. As well known [34], n i=1
qi =
n
μi = 2m.
(1)
i=1
The incidence matrix of G is the n × m matrix I(G) whose (i, j)-entry is 1 if vi is incident to ej and 0 otherwise. It is known that [7] I(G)I(G)t = D(G) + A(G) = Q(G).
(2)
The notion of the energy of a graph G was introduced by one of the present authors in 1978 as the sum of the absolute values of the eigenvalues of A(G). It’s origin was from chemistry, where it is connected with the total π-electron energy of a molecule [24]. Eventually, it was recognized that the interest for this graph invariant goes far beyond chemistry; see the reviews [18,23], the recent monograph [30], and the recent papers [15,39]. For lower and upper bounds on graph energy see [12,13,20,26,30] and the references cited therein. In view of the success of the mathematical theory of graph energy, several energy-like invariants were put forward, based on graph matrices different than A(G). In [32], the so-called Laplacian-energy-like invariant, LEL, was defined as LEL = LEL(G) =
n √
μi .
(3)
i=1
For its basic properties see the survey [31]. Upper and lower bounds on LEL were reported in [4,10,11,14,32,42]. Another energy-like quantity in the incidence energy, IE, invented by Jooyandeh et al. [29]. This graph invariant is defined as the sum of the singular values σi of the
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331
incidence matrix I(G). Recall that the singular values of a matrix M are the nonnegative square roots of MMt or Mt M and that these matrices have the same nonzero eigenvalues. From these facts, and bearing in mind identity (2), it follows that [21,22] IE = IE(G) =
n √
qi .
(4)
i=1
The mathematical theory of incidence energy is nowadays elaborated in considerable detail, see [5,6,21,22,28,33,35,38,41,43,44]. The present paper is organized as follows. In Section 2, we give a list of some previously known results. In Section 3, we present novel lower bounds for IE in terms of n, m, Δ, and clique number. In Section 4, we give novel upper bounds for IE in terms of n, m, independence number, and the first Zagreb index. In Section 5, we deduce some Nordhaus–Gaddum-type results for IE. 2. Preliminaries We state here some previously known results that will be needed in the subsequent three sections. Lemma 2.1 (Jensen’s Inequality). (See [25].) If f is convex on an interval I and x1 , x2 , . . . , xn are in I, then
x1 + x2 + · · · + xn f n
f (x1 ) + f (x2 ) + · · · + f (xn ) . n
For strictly convex functions, equality holds if and only if x1 = x2 = · · · = xn . Lemma 2.2. (See [40].) Let B be a p × p symmetric matrix and let Bk be its leading k × k submatrix. Then, for i = 1, 2, . . . , k, λp−i+1 (B) λk−i+1 (Bk ) λk−i+1 (B)
(5)
where λi (B) is the i-th greatest eigenvalue of B. Lemma 2.3. (See [27].) The Laplacian eigenvalues of G interlace the Laplacian eigenvalues of G + e, i.e., μ1 (G + e) μ1 (G) μ2 (G + e) · · · μn (G + e) μn (G). Lemma 2.4. (See [34].) Let G be a graph with Laplacian spectrum {0 = μn , μn−1 , . . . , μ1 }. Then the Laplacian spectrum of G is {0, n − μ1 , n − μ2 , . . . , n − μn−1 }, where G is the complement of G.
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Lemma 2.5. (See [1,2].) For any α (0 < α 1 or 2 α < 3), α α q1α + q2α + · · · + qnα μα 1 + μ2 + · · · + μn .
Moreover, the equality holds for α ∈ / {1, 2} if and only if G is isomorphic to a bipartite graph. Lemma 2.6. (See [17].) Let G be a simple graph on n vertices with at least one edge. Then q1 Δ + 1. If G is connected, then the equality holds if and only if G ∼ = K1,n−1 . Lemma 2.7. (See [36].) Let G be a simple graph on n vertices with m edges. Then q1 4m/n with equality if and only if G is regular. Lemma 2.8. (See [8].) Let G be a connected graph of order n. Then qn < δ. We denote the set of neighbors of the vertex vi by Ni = {vj ∈ V (G) | vi vj ∈ E(G)}. If G has distinct vertices vi and vj with Ni = Nj , then vi and vj are said to be duplicates and (vi , vj ) is called a duplicate pair [16]. Lemma 2.9. (See [9].) Let G = (V, E) be a graph with vertex set V (G) = {v1 , v2 , . . . , vn }. If G has k − 1 duplicate pairs (vi , vi+1 ), i = 1, 2, . . . , k − 1, then this graph G has at least k − 1 equal signless Laplacian eigenvalues and they are all equal to the cardinality of the neighbor set. The corresponding k − 1 eigenvectors are of the form ( 1, −1, 0, . . . , 0)t , ( 1, 0, −1, 0, . . . , 0)t , . . . , ( 1, 0, . . . , −1, 0, . . . , 0)t . 2
3
k
Lemma 2.10. Let G be a graph with vertex set {v1 , v2 , . . . , vn }. Let G have a clique of size ω whose vertices are v1 , v2 , . . . , vω . If dG (vi ) = ω − 1 for i = 2, 3, . . . , ω, then G has at least ω − 2 mutually equal signless Laplacian eigenvalues, equal to ω − 2. The corresponding ω − 2 eigenvectors are (0, 1, −1, 0, . . . , 0)t , (0, 1, 0, −1, 0, . . . , 0)t , . . . , (0, 1, 0, . . . , −1, 0, . . . , 0)t . 2
3
ω−1
Proof. Let x = (x1 , x2 , . . . , xn )t be an eigenvector of Q(G), corresponding to an eigenvalue q(G). Then for i = 1, 2, . . . , n, q(G)xi = dG (vi )xi +
xj .
j: vi vj ∈E(G)
One can see easily that the eigenvalue ω − 2 with corresponding eigenvectors (0, 1, −1, 0, . . . , 0)t , (0, 1, 0, −1, 0, . . . , 0)t , . . . , (0, 1, 0, . . . , −1, 0, . . . , 0)t 2
3
ω−1
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333
satisfies the above relation. Since these eigenvectors are linearly independent, we conclude that ω − 2 is an eigenvalue of Q(G) with multiplicity at least ω − 2. 2 Lemma 2.11. (See [38].) Let e ∈ E(G). Then, IE(G − e) < IE(G). 3. Lower bounds on incidence energy In this section, we give two lower bounds on IE in terms of n, m, Δ, clique number ω, and the number t of spanning trees. Denote by KS n,ω the graph of order n with clique number ω, obtained by attaching n − ω pendent edges to one vertex of the complete graph Kω . In order to deduce our first lower bound, we need the following results: Lemma 3.1. Let KS n,ω be the graph defined above. Then its signless Laplacian eigenvalues are a, b, ω − 2, ω − 2, . . . , ω − 2, 1, 1, . . . , 1, c ω−2
n−ω−1
where a, b, and c (a b c) are the roots of the equation: x3 − (2ω + n − 3)x2 + n(2ω − 3)x − (ω − 1)(2ω − 4) = 0. Proof. By Lemmas 2.9 and 2.10, we conclude that ω − 2, ω − 2, . . . , ω − 2, 1, 1, . . . , 1 ω−2
n−ω−1
are signless Laplacian eigenvalues of KS n,ω . The remaining eigenvalues satisfy: qx1 = (n − 1)x1 + (ω − 1)x2 + (n − ω)x3 , qx2 = (ω − 1)x2 + (ω − 2)x2 + x1 , qx3 = x3 + x1 . Thus the remaining signless Laplacian eigenvalues of KS n,ω satisfy Eq. (6). 2 Corollary 3.2. Let KS n,ω be the graph defined above. Then IE(KS n,ω ) =
√
a+
√
b+
√
c + (ω − 2)3/2 + n − ω − 1
where a, b, and c are the same as in Lemma 3.1. We are now ready to give a lower bound on IE(G) in terms of n, Δ, and ω.
(6)
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Theorem 3.3. Let G ( Kn ) be a connected graph of order n with maximum degree Δ and clique number ω. Then IE(G)
√
a+
√
b+
√
c + (ω − 2)3/2 + Δ − ω,
(7)
where a, b and c (a b c) are the roots of the equation g(x) = 0, where g(x) = x3 − (2ω + Δ − 2)x2 + (Δ + 1)(2ω − 3)x − (ω − 1)(2ω − 4). Moreover, the equality holds in (7) if and only if G ∼ = KS n,ω . Proof. Since G Kn , we have ω n − 1. We now consider two cases (i) Δ = n − 1, (ii) Δ n − 2. ∼ KS n,ω , then by Case (i): Δ = n − 1. In this case KS n,ω is a subgraph of G. If G = Corollary 3.2, equality holds in (7). Otherwise, |E(G)| > |E(KS n,ω )| and by Lemma 2.11 and Corollary 3.2, IE(G) > IE(KS n,ω ) =
√
a+
√
b+
√
c + (ω − 2)3/2 + Δ − ω.
Case (ii): Δ n − 2. Since ω n − 2, we have Δ ω as G is connected. First we assume that the maximum degree vertex vi is in V (Kω ) of graph G. In this case KS Δ+1,ω is a subgraph of G and |E(G)| > |E(KS Δ+1,ω )| as G is connected. By Lemma 2.11 and Corollary 3.2, we get IE(G) > IE(KS Δ+1,ω ) =
√
a+
√
b+
√
c + (ω − 2)3/2 + Δ − ω.
Assume next that the maximum degree vertex vi is in V (G) \ V (Kω ). Let S be the set of vertices in Kω of G. Then S = V (Kω ) = {v1 , v2 , . . . , vω }. Suppose that NG (vi ) ∩ S = {v1 , v2 , . . . , vp }. Then |NG (vi ) ∩ S| = p. Since ω is the clique number of G, we must have 0 p ω − 1. (For p = ω, the clique number would be ω + 1, a contradiction.) If p = ω − 1, then the ω − 1 = p vertices in S, together with the maximum degree vertex vi , form a clique Kω . The maximum degree vertex vi is in this clique. Hence we get the earlier result (7). Otherwise, 0 p ω − 2. Then Kω ∪ K1,Δ−p is a subgraph of G. The signless Laplacian spectrum of Kω ∪ K1,Δ−p is
2(ω − 1), ω − 2, ω − 2, . . . , ω − 2, Δ − p + 1, 1, 1, . . . , 1, 0, 0 . ω−1
Δ−p−1
Let g(x) be the function defined in Theorem 3.3. Since a, b, and c are the roots g(x) = 0, we have
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a + b + c = 2ω + Δ − 2,
335
(8)
ab + bc + ca = (Δ + 1)(2ω − 3),
(9)
abc = (ω − 1)(2ω − 4). Now, g(0) = −(ω −1)(2ω −4) < 0 and g(1) = 2(ω −2)(Δ−ω +1) > 0. Since a b c, we must have 0 < c < 1. We have to show that √
a+
√
b<
2(ω − 1) +
√
Δ−ω+3+
√
ω−2
(10)
that is, √ a + b + 2 ab < 2ω + Δ − 1 + 2 2(ω − 1)(Δ − ω + 3) + 2(ω − 1)(ω − 2)
+ (ω − 2)(Δ − ω + 3) that is, ab <
1 + 2(ω − 1)(Δ − ω + 3) + 2(ω − 1)(ω − 2) + (ω − 2)(Δ − ω + 3) 4
by (8)
that is, ab < 2(ω − 1)(Δ + 1) which by (9) is always true. Since 0 p ω − 2, we have Δ − p − 1 Δ − ω + 1. Now, IE(G) IE(Kω ∪ K1,Δ−p ) √ = Δ − p + 1 + 2(ω − 1) + (ω − 1) ω − 2 + Δ − p − 1 √ √ Δ − ω + 3 + 2(ω − 1) + (ω − 1) ω − 2 + Δ − ω + 1 √ √ √ √ > a + b + c + (ω − 2) ω − 2 + Δ − ω as c < 1 and by (10). This completes the proof. 2 The following result was obtained in [10]: Lemma 3.4. Let G be a connected graph of order n with m edges, maximum degree Δ, and t spanning trees. Then LEL(G)
√
Δ + 1 + 2m − n + (n − 2)(n − 3)t1/(n−2)
with equality if and only if G ∼ = Kn or G ∼ = K1,n−1 .
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We now establish the incidence-energy analogue of Lemma 3.4: Theorem 3.5. Let G be a connected graph of order n with m edges, maximum degree Δ, and t spanning trees. Then √ IE(G) Δ + 1 + 2m − n + (n − 2)(n − 3)t1/(n−2) (11) with equality holding in (11) if and only if G ∼ = K1,n−1 . Proof. Bearling in mind Eqs. (3) and (4), Lemma 2.5 implies IE(G) =
√
q1 +
√
q2 + · · · +
√
qn
√
μ1 +
√
μ2 + · · · +
√ μn = LEL(G)
(12)
which together with Lemma 3.4 yields (11). Moreover, by Lemma 2.5, equality holds in (12) if and only if G is isomorphic to a bipartite graph. Hence equality holds in (11) if and only if G ∼ = K1,n−1 . 2 4. Upper bounds on incidence energy In this section, we give two upper bounds on IE in terms of n, m, independence number α, and first Zagreb index M1 . The first Zagreb index M1 = M1 (G) is equal to the sum of squares of the vertex degrees of the graph G [19]. Theorem 4.1. Let G be a connected graph of order n with m edges. Then 1/4 16m2 4m 3/4 M1 (G) + 2m − + (n − 1) IE(G) . n n2
(13)
Equality holds if and only if G ∼ = Kn . Proof. We start with n
qi2 =
i=1
n
dG (vi ) dG (vi ) + 1 = M1 + 2m.
i=1
Assume first that G is non-bipartite. Since G is connected, we have qi > 0, i = 1, 2, . . . , n. Then, by applying Lemma 2.1,
1 √ q n − 1 i=2 n
4
1 2 M1 + 2m − q12 qi = n − 1 i=2 n−1 n
that is n √ i=2
1/4 qi (n − 1)3/4 M1 + 2m − q12
(14)
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337
with equality holding if and only if q2 = q3 = · · · = qn . This implies
IE(G) =
√
q1 +
n √
qi
i=2
√
1/4 q1 + (n − 1)3/4 M1 (G) + 2m − q12 .
(15)
Consider now the case when G is bipartite. Since G is connected, qn = 0 and qi > 0, i = 1, 2, . . . , n − 1. This time, instead of (14) we have n √
1/4 qi (n − 2)3/4 M1 (G) + 2m − q12
i=2
1/4 < (n − 1)3/4 M1 (G) + 2m − q12 and we again arrive at the inequality (15). Bearing in mind the bound (15), define an auxiliary function f (x) =
1/4 √ x + (n − 1)3/4 M1 (G) + 2m − x2
whose first derivative is 1 (n − 1)3/4 x f (x) = √ − . 2 x 2(M1 (G) + 2m − x2 )3/4 Since nq12
n
qi2 = M1 (G) + 2m
i=1
one can easily verify that f (x) 0 i.e., that f (x) is a decreasing function. Then by using Lemma 2.7, we get
4m f (x) f n
=
1/4 16m2 4m + (n − 1)3/4 M1 (G) + 2m − n n2
which combined with (15) yields (13). First part of the proof is done. Suppose now that equality holds in (13). Then all inequalities in the above argument must be equalities. This requires that q2 = q3 = · · · = qn . Moreover, it also must be q1 = 4m/n and thus by Lemma 2.7 G must be regular.
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By contradiction, we show that G is isomorphic to Kn . For this suppose that G is a Δ-regular graph, Δ = n − 1. Then there exist two vertices vi and vj in G that are not adjacent. By Lemma 2.2, q1 q1 and q2 q2 , where q1 and q2 are the solutions of Δ−q 0 = 0. 0 Δ−q Therefore, q1 Δ and q2 Δ. By Lemma 2.8, qn < Δ, as G is a regular graph different from Kn . Then, however, Δ q2 = qn < Δ, which is a contradiction. Therefore, it must be G ∼ = Kn . Conversely, let G ∼ = Kn . Then
1/4 √ 16m2 4m 3/4 M1 (G) + 2m − + (n − 1) = 2(n − 1) + (n − 1) n − 2 = IE(G). n n2
This completes the proof. 2 Corollary 4.2. Let G be a connected graph of order n with m edges and maximum degree Δ. Then IE(G) <
√
1/4 Δ + 1 + (n − 1)3/4 M1 (G) + 2m − (Δ + 1)2 .
(16)
Proof. The function f (x) in Theorem 4.1 is decreasing. By Lemma 2.6, q1 Δ + 1. Using these results, we get (16). Suppose now that the equality holds in (16). Moreover, from Theorem 4.1 we know that q1 = Δ + 1 and q2 = q3 = · · · = qn . Since G is connected, by Lemma 2.6, we have G ∼ = K1,n−1 and hence 1 = q2 = qn = 0, a contradiction. This completes the proof. 2 A complete split graph CS(n, ω), ω n, is a graph on n vertices consisting of a clique on ω vertices and a stable set on the remaining n − ω vertices, such that each vertex of the clique is adjacent to each vertex of the stable set. Given a graph G, a subset S of V (G) is called an independent set of G if G[S], the subgraph induced by S, is a graph with |S| isolated vertices. The independence number of G is denoted by α and is defined to be the number of vertices in the largest independent set of G. The signless Laplacian eigenvalues of the complete split graph CS(n, n − α), 1 α n − 1, are [9]:
1 3n − 2α − 2 ± (n − 2)2 + 4α(n − α) , n − 2, . . . , n − 2, n − α, . . . , n − α, 2 n−α−1
α−1
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from which it straightforwardly follows √ √ IE CS(n, n − α) = (n − α − 1) n − 2 + (α − 1) n − α 1 +√ 3n − 2α − 2 + (n − 2)2 + 4α(n − α) 2 1 +√ 3n − 2α − 2 − (n − 2)2 + 4α(n − α). 2 Theorem 4.3. Let G be a graph of order n with independence number α. Then IE(G) IE CS(n, n − α) with equality if and only if G ∼ = CS(n, n − α). Proof. By Lemma 2.11, if a new edge is inserted to the graph G, then its independence energy will increase. Adding to G edges one-by-one, so that its α-value is not changed, we arrive at CS(n, n − α). Thus, if G CS(n, n − α), then IE(G) < IE(CS(n, n − α)). 2 5. Nordhaus–Gaddum-type results for incidence energy Nordhaus and Gaddum [37] gave bounds for the sum of the chromatic numbers of a graph and its complement. Eventually, numerous Nordhaus–Gaddum-type results for other graph invariants were obtained [3]. With regard to incidence energy, the following result is known [22]: Lemma 5.1. Let G be a graph of order n 2. Then √ √ (n − 1) n IE(G) + IE(G) < 2 n − 1 + (n − 1) 2(n − 2) with left equality if and only if n = 2. Here we give a few more Nordhaus–Gaddum-type results for IE. First we give a lower bound on IE(G) + IE(G) in terms of n and the number of spanning trees t and t for G and G, respectively. Theorem 5.2. Let G be a graph of order n 2. Then 1/[2(n−1)] IE(G) + IE(G) (n − 1) nn−1 + n2n−1 tt .
(17)
Equality holds if and only if G ∼ = K2 . Proof. For n = 2, one can see easily that the inequality holds in (17) and that equality holds in (17) for G ∼ = K2 . Let, therefore, n 3. We have μi 0 and n − μi 0, that is μi (n − μi ) 0, i = 1, 2, . . . , n − 1.
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One can see easily that
n + 2 μ1 (n − μ1 ) n + 2 μ2 (n − μ2 ) · · · n + 2 μn−1 (n − μn−1 ) n−1 n−1 n−1 n−1 n +2 μi × (n − μi ). i=1
i=1
Since t = t(G) =
n−1 1 μi n i=1
and t = t(G) =
n−1 1 (n − μi ) n i=1
from the above, we get n−1
n + 2 μi (n − μi ) nn−1 + 2n−1 n2 tt.
(18)
i=1
By the arithmetic–geometric mean (AGM) inequality,
n−1 1 n + 2 μi (n − μi ) n − 1 i=1
n−1
n−1
n + 2 μi (n − μi )
1/2 .
(19)
i=1
Now, IE(G) + IE(G) =
n √ ( qi + qi ) i=1
n √ ( μi + μi )
by Lemma 2.5
(20)
i=1
=
n−1
√ √ ( μi + n − μi ) by Lemma 2.4
i=1
=
n−1
n + 2 μi (n − μi )
i=1
n−1 1/[2(n−1)] (n − 1) n + 2 μi (n − μi )
by (19).
i=1
Using (18), we arrive at inequality (17). Suppose now that equality holds in (17). Then all inequalities in the above argument must be equalities. From the equality in (20), by Lemma 2.5, we get that both G and G are bipartite. From the equality in (19), it follows μ1 (n − μ1 ) = μ2 (n − μ2 ) = · · · = μn−1 (n − μn−1 )
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that is, μi (n − μi ) = μj (n − μj ) that is, (μi − μj )(μi + μj − n) = 0 for 1 i < j n. This implies that either μi = μj or μi + μj = n. Therefore, G must be isomorphic to a graph H such that μ1 (H) = μ2 (H) = · · · = μp (H) and μp+1 (H) = μp+2 (H) = · · · = μn−1 (H) = n − μ1 (H). Consequently, G is bipartite. From the equality in (18), we get
p n−p−1 n + 2 μ1 (H) n − μ1 (H) × n + 2 μn−1 (H) n − μn−1 (H) p n−p−1 n−p−1 = nn−1 + 2n−1 μp1 (H)μn−1 (H) n − μ1 (H) n − μn−1 (H)
that is, n−1 n + 2 μ1 (H)μn−1 (H) = nn−1 + 2n−1 μn−1 (H)μn−1 1 n−1 (H). Since n 3, one can see easily that n−1 n + 2 μ1 (H)μn−1 (H) > nn−1 + 2n−1 μn−1 (H)μn−1 1 n−1 (H). This completes the proof. 2 Remark 5.3. The lower bound (17) is always better than the lower bound in Lemma 5.1. We now give an upper bound on IE(G) + IE(G) in terms of n, m, Δ, and δ. Theorem 5.4. Let G be a graph of order n 2, with m edges, maximum degree Δ and minimum degree δ. Then IE(G) + IE(G)
√
2Δ + 2(n − δ − 1)
(n − 1)(n − 2) √ 2m + n(n − 1) − 2m + n
with equality if and only if G ∼ = Kn .
(21)
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Proof. By the Cauchy–Schwarz inequality,
IE(G) + IE(G) =
√
q1 +
q1 +
n √
qi +
i=2
√
2Δ +
n
qi
i=2
n n 2(n − δ − 1) + (n − 1) qi + (n − 1) qi i=2
(22)
i=2
as q1 2Δ and q 1 2(n − δ − 1) √ = 2Δ + 2(n − δ − 1) + (n − 1)(2m − q1 ) + (n − 1)(2m − q 1 ) as
n
qi = 2m and
i=1
n
q i = 2m
i=1
√ 2Δ + 2(n − δ − 1) +
4m 4m + (n − 1) 2m − (n − 1) 2m − n n
(23)
4m 4m and q 1 , by Lemma 2.7 n n √
(n − 1)(n − 2) √ = 2Δ + 2(n − δ − 1) + 2m + n(n − 1) − 2m n as q1
as 2m + 2m = n(n − 1). The first part of the proof is done. Suppose now that equality holds in (21). Then all inequalities in the above argument must be equalities. From the equality in (22), we get q1 = 2Δ, q1 = 2(n − δ − 1), q2 = q3 = · · · = qn , and q 2 = q 3 = · · · = q n . From the equality in (23), we get q1 = 4m/n and q1 = 4m/n. By Lemma 2.7, this implies that G and G are isomorphic to a regular graph. Assume first that G is connected. In a similar manner as in the proof of Theorem 4.1, we get G ∼ = Kn . If G is disconnected, then G is connected. Moreover, we have q 2 = q 3 = · · · = q n . As before, we can easily show that G ∼ = Kn . Hence G ∼ = K n. Conversely, one can see easily that the equality holds in (21) for complete graph Kn or for null graph K n . 2 Remark 5.5. For the complete graph Kn , the upper bound (21) is better than the upper bound in Lemma 5.1.
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