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Proof for four conjectures about the distance Laplacian and distance signless Laplacian eigenvalues of a graph
Linear Algebra and its Applications 471 (2015) 10–20
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Proof for four conjectures about the distance Laplacian and distance signless Laplacian eigenvalues of a graph Fenglei Tian, Dein Wong ∗,1 , Jianling Rou Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, PR China
a r t i c l e
i n f o
Article history: Received 7 April 2014 Accepted 15 December 2014 Available online xxxx Submitted by S. Kirkland MSC: 05C50 15A18 Keywords: Distance Laplacian eigenvalues Distance matrix Unicyclic graphs and trees
a b s t r a c t The distance Laplacian matrix L(G) of a graph G is defined to be L(G) = diag(Tr) − D(G), where D(G) denotes the distance matrix of G and diag(Tr) denotes the diagonal matrix of the vertex transmissions in G. Similarly, the distance signless Laplacian matrix of G is defined as Q(G) = diag(Tr) +D(G). The eigenvalues of L(G) and Q(G) are called the distance Laplacian and distance signless Laplacian eigenvalues, respectively. In this paper, four conjectures proposed by M. Aouchche and P. Hansen about the largest and the second largest distance Laplacian eigenvalues and the second largest distance signless Laplacian eigenvalue of a graph are proved. © 2014 Elsevier Inc. All rights reserved.
1. Introduction Throughout this paper, all graphs considered are simple, undirected and connected. A graph is denoted by G = G(V, E) with vertex set V (G) and edge set E(G). The * Corresponding author. 1
E-mail address: [email protected] (D. Wong). Supported by the National Natural Science Foundation of China (11171343).
http://dx.doi.org/10.1016/j.laa.2014.12.015 0024-3795/© 2014 Elsevier Inc. All rights reserved.
F. Tian et al. / Linear Algebra and its Applications 471 (2015) 10–20
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order of G is defined as the cardinality of the set V (G). The set of vertices adjacent to u ∈ V (G), denoted by N (u), refers to the neighborhood of u, and the degree of u means the cardinality of N (u). A pendent vertex is a vertex with degree one. The symbol u ∼ v means that two vertices u and v of G are adjacent. u and v are called twin points if NG (u) = NG (v) for u, v ∈ V (G). Generally, letting S ⊆ V (G), if NG (u) = NG (v) for any u, v ∈ S, we call S a set of twin points. If G is a tree, then a set of twin points consists of pendent vertices. The distance between u ∈ V (G) and v ∈ V (G), denoted by d(u, v), is the length of a shortest path between u and v. In particular, d(u, u) = 0 for any vertex u ∈ V (G). The diameter of G, denoted by d(G), is the maximum distance between vertices. The transmission Tr G (u) of a vertex u in G is defined as the sum of distances of u to all vertices of V (G), i.e., Tr G (u) = v∈V (G) d(v, u). The Wiener index W (G) of a graph G is defined as W (G) = 12 u,v∈V d(u, v). Obviously, W (G) = 12 v∈V Tr(v). Usually, we use Kn , Cn , Pn and Sn to denote the complete graph, the cycle, the path and the star with order n, respectively. By Sn+ we denote the graph obtained from Sn by adding an edge joining two pendent vertices of Sn . The complete bipartite graph with two color classes of order a and b is denoted by Ka,b . We denote by G − u, for u ∈ V (G), the induced subgraph of G obtained by deleting the vertex u and all the edges incident to it, and by G − e, for e ∈ E(G), we denote the subgraph of G obtained by deleting e from G. A subgraph H of G is called a spanning subgraph if V (H) = V (G), i.e., H is obtained from G by deleting some edges from G. Suppose that V (G) = {v1 , v2 , . . . , vn }. The distance matrix of G, denoted by D(G), is an n × n real symmetric matrix with the (i, j)-entry being d(vi , vj ). Obviously, the transmission Tr G (v) of a vertex v is the sum of all coordinates of the row vector of D(G) indexed by v. For a graph G, denote by diag(Tr) the diagonal matrix with the (i, i)-entry being Tr G (vi ). Distance Laplacian matrix introduced by M. Aouchiche and P. Hansen [1], is defined as L(G) = diag(Tr) −D(G). The eigenvalues of L(G), written as {∂1 (G), ∂2 (G), . . . , ∂n (G)}, are called the distance Laplacian eigenvalues of G. Sometimes we write the distance Laplacian eigenvalues of G as {∂1 , ∂2 , . . . , ∂n } for short, and labeled them such that ∂1 ≥ ∂2 ≥ . . . ≥ ∂n . Distance signless Laplacian matrix Q(G) is defined as Q(G) = diag(Tr) + D(G). Similarly, the distance eigenvalues and the distance signless Laplacian eigenvalues of a graph G of order n are respectively labeled as δ1 (G) ≥ δ2 (G) ≥ . . . ≥ δn (G); q1 (G) ≥ q2 (G) ≥ . . . ≥ qn (G). A real matrix A with nonnegative entries is called irreducible if for all i, j there is a k such that the (i, j)-entry of Ak is positive. For an n × n real symmetric matrix A, we denote by λ(A) one of its eigenvalues. Usually, the eigenvalues of A are labeled as λ1 (A) ≥ λ2 (A) ≥ . . . ≥ λn (A). Distance spectrum and distance signless Laplacian spectrum of graphs have attracted a lot of attention (see [3–10]). However, little is known for the distance Laplacian
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spectrum of graphs. In [1], M. Aouchiche and P. Hansen gave the distance Laplacian characteristic polynomials of some special graphs, and proved that the distance Laplacian eigenvalues do not decrease on deletion of edges. In [2], the authors introduced some properties for the distance Laplacian eigenvalues of graphs. Especially, they proposed nine conjectures on the largest and the second largest eigenvalue of the distance Laplacian matrix of a graph. This paper is devoted to prove three of the conjectures in [2] and a conjecture in [12], listed as follows. Conjectures (proposed by M. Aouchiche and P. Hansen in [2,12]). • Let G be a unicyclic graph with order n (≥ 6), then ∂1 (G) ≥ ∂1 (Sn+ ) with equality if and only if G = Sn+ . • For any graph G on n (≥ 4) vertices, ∂2 (G) ≥ n with equality if and only if G = Kn , or G = Kn − e for some e ∈ E(Kn ). • If T is a tree with order n (≥ 5), then ∂2 (T ) ≥ 2n − 1 with equality if and only if T = Sn . • Let T be a tree on n (≥ 4) vertices and q2 be the second largest eigenvalue of Q(T ). Then q2 ≥ 2n − 5, with equality if and only if T = Sn . 2. Preliminaries and lemmas Before giving the proof of the conjectures, we introduce some known conclusions used in the next section. At the end of this section, some plain however interesting results are given. Lemma 2.1 (Courant–Weyl inequality). (See [11, p. 31].) Let A1 and A2 be real symmetric matrices of order n. Then λn (A2 ) + λi (A1 ) ≤ λi (A1 + A2 )
for 1 ≤ i ≤ n.
Lemma 2.2 (Interlacing theorem). (See [11, p. 30].) Let A be a real symmetric matrix of order n and M be a principal submatrix of A with order s (≤ n). Then λi+n−s (A) ≤ λi (M ) ≤ λi (A),
1 ≤ i ≤ s.
Lemma 2.3 (Perron–Frobenius theorem). Let S = (sij ) be a complex matrix, and A be an irreducible matrix of the same order. |S| denotes the matrix whose (i, j)-entry is |sij |. If |S| ≤ A and S has σ as an eigenvalue, then |σ| ≤ λ1 (A). If the equality holds, then |S| = A, and there is a diagonal matrix E with diagonal entries of absolute value 1 and a constant c of absolute value 1, such that S = cEAE −1 .
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Lemma 2.4 (Gershgorin theorem). (See [2].) Let M = (mij ) be a complex n × n matrix and denote by λ1 , λ2 , . . . , λp its distinct eigenvalues. Then {λ1 , λ2 , . . . , λp } ⊂
n
z : |z − mii | ≤
i=1
|mij | .
j=i
Lemma 2.5. (See [2, Theorem 2.2].) For any graph G, ∂n (G) = 0 with multiplicity 1. Lemma 2.6. (See [1, Corollary 3.6].) Let G be a graph on n (≥ 3) vertices. Then qi (G) ≥ n − 2 for 2 ≤ i ≤ n. Lemma 2.7. (See [2, Theorem 2.3].) Let G be a graph of order n. Let S be a set of twin points of G, where S = {v1 , v2 , . . . , vk } and 1 < k < n. Then Tr(vi ) + 2 is an eigenvalue of L(G) with multiplicity at least k − 1. Lemma 2.8. (See [2, Theorem 2.7].) If G is a graph on n (≥ 2) vertices, then the multiplicity of the largest eigenvalue ∂1 of L(G) satisfies m(∂1 ) ≤ n − 1, with equality if and only if G = Kn . Lemma 2.9. (See [4, Lemma 2.3].) Let G be a unicyclic graph of order n (≥ 6), and G = Sn+ . Then W (G) ≥ n2 − n − 4 > W Sn+ . Lemma 2.10. (See [1, Theorem 3.5].) Let G be a graph on n vertices and with m (≥ n) ≥ ∂i (G) Then ∂i (G) edges. Denote the graph obtained by deleting an edge from G by G. for i = 1, 2, . . . , n. In the end of this section we give some plain results. For a graph G of order n, obviously, L(G) is a real symmetric matrix. Then the following proposition follows from Lemma 2.2 immediately. Proposition 2.11. Let G be a graph with order n, and let M be the principal submatrix of L(G) with order s ≤ n. Then ∂i+n−s ≤ λi (M ) ≤ ∂i ,
1 ≤ i ≤ s.
2
Let Tr max (G) be the maximal vertex transmission of a graph G. Then the following result can be easily derived from Lemma 2.2 and Lemma 2.4. Proposition 2.12. Tr max (G) ≤ ∂1 ≤ 2Tr max (G). Proof. Suppose v ∈ V (G) such that Tr G (v) = Tr max (G). For the left inequality, let A be a 1 × 1 matrix with the only entry being Tr G (v). Then A is a principal submatrix
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of L(G). Applying Lemma 2.2, we have Tr max (G) = λ1 (A) ≤ λ1 (L(G)) = ∂1 . The right inequality is a direct conclusion of Gershgorin theorem. 2 Proposition 2.13. Let G be a graph of order n (≥ 3). Then (a) 2δi (G) ≤ qi (G), 1 ≤ i ≤ n; (b) q1 (G) > ∂1 . Proof. Obviously, Q(G) = L(G) + 2D(G). Lemma 2.5 says that L(G) is positive semidefinite with 0 being the least eigenvalue, i.e., ∂n = 0. Applying Lemma 2.1, we have that qi (G) = λi (Q(G)) ≥ λi (2D(G)) + λn (L(G)) = 2δi (G) + 0 = 2δi (G). Obviously, |L(G)| = Q(G), and Q(G) is irreducible (see [4]). Applying Lemma 2.3, we have ∂1 = |∂1 | ≤ q1 (G). Here, the equality fails to hold, since the least eigenvalue of L(G) is 0 (using Lemma 2.5) and the least eigenvalue of Q(G) is at least n − 2 (using Lemma 2.6), forcing L(G) = cEQ(G)E −1 for any c with |c| = 1 and any invertible matrix E. 2 Remark 2.14. By the left inequality of Proposition 2.12 and by (b) of Proposition 2.13, we obtain a strict inequality q1 (G) > Tr max (G), which improves Lemma 2.3 in [3]. 3. Proof for the four conjectures In this section, we demonstrate the conjectures given at the end of Section 1. Theorem 3.1. Let G be a unicyclic graph with order n (≥ 6), then ∂1 (G) ≥ ∂1 (Sn+ ) with equality if and only if G = Sn+ . Proof. The explanation for Sn+ in [1] tells us the largest distance Laplacian eigenvalue of Sn+ with n > 3 is 2n − 1. Thus, to complete the proof we only need to prove that ∂1 (G) > ∂1 (Sn+ ) if G is a unicyclic graph of order n (≥ 6) which is different from Sn+ . If this is n the case, then by Lemma 2.9, W (G) ≥ n2 − n − 4 > W (Sn+ ). Since i=1 ∂i = 2W (G), it follows from Lemma 2.5 and Lemma 2.8 that (n − 1)∂1 > 2W (G) (also see [2]). Thus, we have ∂1 >
2(n2 − n − 4) 8 2W (G) ≥ = 2n − . n−1 n−1 n−1
Consequently, if n ≥ 9, then ∂1 > 2n −
8 ≥ 2n − 1. n−1
If 6 ≤ n ≤ 8, by a direct calculation we also obtain ∂1 > 2n − 1. Therefore, ∂1 (G) ≥ 2n − 1 = ∂1 (Sn+ ), and the equality holds if and only if G = Sn+ . 2
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Theorem 3.2. Let G be a graph on n (≥ 4) vertices. Then ∂2 (G) ≥ n, with equality if and only if G = Kn or G = Kn − e with e ∈ E(Kn ).
n be the spanning subgraph of Kn obtained by deleting two different edges, Proof. Let K
n ) > n. say e1 , e2 , from Kn . Firstly, we prove that ∂2 (K Case 1. e1 and e2 have a common vertex. Suppose the common vertex of e1 and e2 is v1 and the other two end vertices are v2
n ), written as M , indexed by v1 , v2 and v3 . Considering the principal submatrix of L(K and v3 , we have ⎛
n + 1 −2 ⎝ M= −2 n −2 −1
⎛ ⎞ −2 1 ⎝ ⎠ −1 = nI3 + −2 n −2
⎞ −2 −2 0 −1 ⎠ . −1 0
The second largest eigenvalue of M is λ2 (M ) = n + 1. By Proposition 2.11, we have
n ) ≥ λ2 (M ) = n + 1 > n. ∂2 (K Case 2. e1 and e2 have no common vertices. Let the end vertices of e1 and e2 be v1 , v2 and v3 , v4 , respectively. Let the principal
n indexed by v1 , v2 , v3 , v4 be M1 . Then, submatrix of K ⎛
n −2 −1 ⎜ −2 n −1 M1 = ⎜ ⎝ −1 −1 n −1 −1 −2
⎛ ⎞ −1 0 −2 −1 ⎜ −2 0 −1 −1 ⎟ ⎟ = nI4 + ⎜ ⎝ −1 −1 0 −2 ⎠ n −1 −1 −2
⎞ −1 −1 ⎟ ⎟. −2 ⎠ 0
The second largest eigenvalue of M1 is λ2 (M1 ) = n + 2. Applying Proposition 2.11 again, we have
n ) ≥ λ2 (M1 ) = n + 2 > n. ∂2 (K Let G be a graph of order n. If G = Kn and G = Kn − e, then it can be viewed
n by deleting some edges. Applying Lemma 2.10, we have as a graph obtained from K
n ) > n. On the other hand, if G = Kn or G = Kn − e, from 1.1, 1.2 in ∂2 (G) ≥ ∂2 (K [1] it follows that ∂2 (G) = n. Finally, we have that ∂2 (G) ≥ n, and the equality holds if and only if G = Kn or G = Kn − e with e ∈ E(Kn ). 2 The following two lemmas will be applied in the proof of the third conjecture. Let T be a tree with diameter d. Suppose P = v1 v2 . . . vd+1 is a longest path in T such that vi ∼ vi+1 for i = 1, 2, . . . , d, then P is a diameter of tree T . For each vi ∈ V (P ), denote by Pvi the induced connected subgraph of T with maximum possible vertices, which contains the vertex vi and contains no other vertices of P . Now we construct a
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Fig. 1. The example of T and T .
new tree (with the same order and diameter as T ) from T in the following way: for each vi ∈ V (P ), we delete all the edges in Pvi and add edges joining vi . The resultant tree is denoted by T (see Fig. 1 for example). Lemma 3.3. Let T , T and P be as described above. Then max Tr T (v1 ), Tr T (vd+1 ) ≥ max Tr T (v1 ), Tr T (vd+1 ) , where d = d(T ), and v1 , vd+1 are the end vertices of P . / V (P ), then Proof. For v ∈ V (T ), if v ∈ V (P ), then dT (v1 , v) = dT (v1 , v); if v ∈ dT (v1 , v) ≥ dT (v1 , v). Consequently,
Tr T (v1 ) =
dT (v1 , v) ≥
v∈V (T )
v∈V
dT (v1 , v) = Tr T (v1 ).
(T )
Similarly, Tr T (vd+1 ) ≥ Tr T (vd+1 ). This completes the proof. 2 Lemma 3.4. Let T be a tree with diameter d. Then Tr max (T ) ≥
1 n(d + 2) − (d + 1). 2
Proof. Let P = v1 v2 . . . vd+1 and T be defined as above. Clearly, in T , for a vertex v outside P , d(v1 , v) + d(vd+1 , v) = d + 2, and for a vertex v of P , d(v1 , v) + d(vd+1 , v) = d. Thus, we have Tr T (v1 ) + Tr T (vd+1 ) = (n − d − 1)(d + 2) + (d + 1)d = n(d + 2) − 2(d + 1), from which it follows that
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1 max Tr T (v1 ), Tr T (vd+1 ) ≥ Tr T (v1 ) + Tr T (vd+1 ) 2 1 = n(d + 2) − (d + 1). 2 Applying Lemma 3.3, we have Tr max (T ) ≥ max Tr T (v1 ), Tr T (vd+1 ) ≥ max Tr T (v1 ), Tr T (vd+1 ) ≥
1 n(d + 2) − (d + 1). 2
2
Remark 3.5. By the proof of Lemma 3.4, one can easily see that max{Tr T (v1 ), Tr T (vd+1 )} ≥ 12 n(d + 2) − (d + 1), where v1 , vd+1 are the end vertices of a longest path and d = d(T ). Now we are in a position to prove the third conjecture. Theorem 3.6. Let T be a tree with order n (≥ 5). The second largest Laplacian eigenvalue ∂2 (T ) of T satisfies ∂2 (T ) ≥ 2n − 1, and the equality holds if and only if T = Sn . Proof. Let T be a tree with diameter d. Suppose P = v1 v2 . . . vd+1 is a longest path in G such that vi ∼ vi+1 for i = 1, 2, . . . , d. Suppose the set of twin points containing v1 (resp., vd+1 ) is W1 (resp., Wd+1 ). Let n1 = |W1 | and nd+1 = |Wd+1 |. The description of 1.3 in [1] tells us the second largest distance Laplacian eigenvalue of Sn is 2n − 1. Thus, to complete the proof we only need to prove that ∂2 (T ) > 2n − 1 if T = Sn and T has at least 5 vertices. If T = Sn , then d ≥ 3. We complete the proof by considering three different cases for d. Case 1. d = 3. Assume n1 or nd+1 is more than 2. Suppose n1 > 2, without loss of generality. Then, it follows from Lemma 2.7 that T has Tr T (v1 ) + 2 as its distance Laplacian eigenvalue with multiplicity at least n1 − 1 (≥ 2). Obviously, Tr T (v1 ) = d(v1 , v1 ) + d(v1 , v2 ) + d(v1 , v4 ) + ≥0+1+3+
d(v, v1 )
v ∈{v / 1 ,v2 ,v4 }
2 = 4 + 2(n − 3) = 2n − 2.
v ∈{v / 1 ,v2 ,v4 }
Thus we have ∂2 (T ) ≥ Tr T (v1 ) + 2 ≥ 2n > 2n − 1. If n1 = 2 and nd+1 = 2, from Lemma 2.7, T has Tr T (v1 ) + 2 and Tr T (vd+1 ) + 2 as its distance Laplacian eigenvalues with multiplicity at least n1 − 1 (= 1) and at least nd+1 − 1 (= 1), respectively. Similar as the case that n1 > 2, Tr T (v1 ) ≥ 2n − 2 and Tr T (vd+1 ) ≥ 2n − 2, from which we have ∂2 (T ) ≥ 2n > 2n − 1.
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Fig. 2. The graphs G1 , G2 , . . . , G8 .
For the left subcases of d = 3, T has five vertices (see G1 in Fig. 2). By a direct calculation we confirm that ∂2 (T ) > 2n − 1. Case 2. d = 4. If n1 > 2, or nd+1 > 2, or n1 = nd+1 = 2, by similar discussions as Case 1, we have ∂2 (T ) > 2n − 1. The parallel discussions are omitted. If n1 + nd+1 = 3, say n1 = 2, nd+1 = 1, without loss of generality. Assume the principal submatrix of L(T ) indexed by Tr T (v1 ) −2 v1 and its twin point, say v1 , is M0 = . It is easy to see that λ2 (M0 ) = −2 Tr T (v1 ) Tr T (v1 ) − 2. Clearly, 5 Tr T (v1 ) = d v1 , v1 + d(v1 , vi ) + i=1
d(v1 , v)
v ∈{v / i ,v1 :1≤i≤5,}
≥ 2 + 10 + 3(n − 6) = 3n − 6. Consequently, we have λ2 (M0 ) = Tr T (v1 ) − 2 ≥ 3n − 8. If n > 7, then λ2 (M0 ) > 2n − 1. So, ∂2 (T ) ≥ λ2 (M0 ) > 2n − 1. When n = 6 or n = 7 in this subcase, we confirm the conclusion by a direct calculation. In the remainder of this case we may assume that each end vertex of every longest path has no other twin points other than itself (see G2 in Fig. 2). Let M be the principal Tr T (v1 ) −4 submatrix indexed by v1 , v5 of G2 . Then M = . Obviously, Tr T (v1 ) = −4 Tr T (v5 ) Tr T (v5 ), and Tr T (v1 ) = d(v1 , v1 ) + d(v1 , v2 ) + d(v1 , v3 ) + d(v1 , v5 ) + ≥0+1+2+4+
d(v, v1 )
v ∈{v / 1 ,v2 ,v3 ,v5 }
3 = 7 + 3(n − 4) = 3n − 5.
v ∈{v / 1 ,v2 ,v3 ,v5 }
Applying Proposition 2.11, we have ∂2 (T ) ≥ λ2 (M ) = Tr T (v1 ) − 4, from which it follows that ∂2 (T ) ≥ Tr T (v1 ) − 4 ≥ 3n − 9. If n ≥ 9, then ∂2 (T ) > 2n − 1. If 5 ≤ n ≤ 8 (see G3 , . . . , G8 in Fig. 2), the conclusion is verified by a direct calculation.
F. Tian et al. / Linear Algebra and its Applications 471 (2015) 10–20
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Case 3. d ≥ 5. Without loss of generality, we suppose that Tr T (v1 ) ≥ Tr T (vd+1 ). As v2 is adjacent to v1 , Tr T (v1 ) = Tr T (v2 ) + n − 2. Let M1 be the principal submatrix of L(T ) indexed by v1 and v2 . Then M1 =
−1 Tr T (v1 ) −1 Tr T (v2 )
= Tr T (v1 )I2 +
0 −1 −1 2 − n
.
The second largest eigenvalue of M1 is λ2 (M1 ) = Tr T (v1 ) − Remark 3.5 tells us that Tr T (v1 ) ≥ (n − 2)2 + 4 ≤ n. Hence,
1 (n − 2) + (n − 2)2 + 4 . 2 1 2 n(d
+ 2) − (d + 1), and it is obvious that
1 n(d + 2) − (d + 1) − (n − 1) λ2 (M1 ) ≥ 2
=
1 nd − d. 2
Applying Proposition 2.11, we obtain ∂2 (T ) ≥ λ2 (M1 ) ≥ 12 nd − d. If n > 8, it is clear that ∂2 (T ) ≥ 12 nd − d > 2n − 1 for d ≥ 5. If 6 ≤ n ≤ 8, we confirm ∂2 (T ) > 2n − 1 by a direct calculation. Finally, we have ∂2 (T ) ≥ ∂2 (Sn ) = 2n − 1, and the equality holds if and only if T = Sn . 2 Ultimately, we prove the last conjecture first proposed in [12]. Theorem 3.7. Let T be a tree on n (≥ 4) vertices and q2 be the second largest eigenvalue of Q(T ). Then q2 ≥ 2n − 5, with equality if and only if T = Sn . Proof. As in Theorem 3.6, let the diameter of T be d. Suppose P = v1 v2 . . . vd+1 is a longest path in G such that vi ∼ vi+1 for i = 1, 2, . . . , d. Suppose the set of twin points containing v1 (resp., vd+1 ) is W1 (resp., Wd+1 ). Let n1 = |W1 | and nd+1 = |Wd+1 |. Section 1.3 in [1] tells us that q2 (Sn ) = 2n − 5 if n ≥ 4. To complete the proof it suffices to prove that q2 (T ) > 2n − 5 if T = Sn and n ≥ 4. Case 1. d = 3. If n = 4, then T = P = P4 . By a direct calculation, we obtain the result. If n > 4, then one of the end vertices of P , say v1 , has a twin point, say v1 , which is different from v1 . Therefore, Tr T (v1 ) = Tr T v1 ≥ 2(n − 4) + 1 + 2 + 3 = 2n − 2.
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Tr T (v1 ) 2 Let the principal submatrix of Q(T ), indexed by v1 and v1 , be M = . 2 Tr T (v1 ) The second largest eigenvalue of M is λ2 (M ) = Tr T (v1 ) − 2. Thus, by Proposition 2.11, we have q2 ≥ λ2 (M ) = Tr T (v1 ) − 2 ≥ 2n − 4 > 2n − 5. Case 2. d ≥ 4. Similar as Case 3 of Theorem 3.6, the principal submatrix of Q(T ) indexed by v1 and v2 , is M1 =
Tr T (v1 ) 1
1 Tr T (v2 )
= Tr T (v1 )I2 +
0 1 1 2−n
,
since Tr T (v1 ) = Tr T (v2 ) + n − 2. For the second largest eigenvalue of M1 , 1 (n − 2) + (n − 2)2 + 4 2 1 ≥ Tr T (v1 ) − n + 1 ≥ nd − d. 2
λ2 (M1 ) = Tr T (v1 ) −
So, we have q2 ≥ λ2 (M1 ) ≥ 12 nd − d. If n > 4, then q2 ≥ 12 nd − d > 2n − 5, as desired. Finally, we have that q2 > 2n − 5 if n ≥ 4, and the equality holds if and only if T = Sn . 2 References [1] M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013) 21–33. [2] M. Aouchiche, P. Hansen, Some properties of the distance Laplacian eigenvalues of a graph, Les Cahiers du GERAD, G-2013-28, http://www.gerad.ca/fichiers/cahiers/G-2013-28.pdf. [3] Rundan Xing, Bo Zhou, On the distance and distance signless Laplacian spectral radii of bicyclic graphs, Linear Algebra Appl. 439 (2013) 3955–3963. [4] R. Xing, B. Zhou, J. Li, On the distance signless Laplacian spectral radius of graphs, Linear Multilinear Algebra 62 (2014) 1377–1387. [5] A. Ilić, Distance spectral radius of trees with given matching number, Discrete Appl. Math. 158 (2010) 1799–1806. [6] M. Nath, S. Paul, On the distance spectral radius of bipartite graphs, Linear Algebra Appl. 436 (2012) 1285–1296. [7] D. Stevanović, A. Ilić, Distance spectral radius of trees with fixed maximum degree, Electron. J. Linear Algebra 20 (2010) 168–179. [8] S.S. Bose, M. Nath, S. Paul, On the maximal distance spectral radius of graphs without a pendent vertex, Linear Algebra Appl. 438 (2013) 4260–4278. [9] S.N. Ruzieh, D.L. Powers, The distance spectrum of the path Pn and the first distance eigenvector of connected graphs, Linear Multilinear Algebra 28 (1990) 75–81. [10] G. Yu, On the least distance eigenvalue of a graph, Linear Algebra Appl. 439 (8) (2013) 2428–2433. [11] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, New York, 2012. [12] M. Aouchiche, P. Hansen, A signless Laplacian for the distance matrix of a graph, Les Cahiers du GERAD, G-2011-78, http://www.gerad.ca/fichiers/cahiers/G-2011-78.pdf.
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Proof for four conjectures about the distance Laplacian and distance signless Laplacian eigenvalues of a graph Fenglei Tian, Dein Wong ∗,1 , Jianling Rou Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, PR China
a r t i c l e
i n f o
Article history: Received 7 April 2014 Accepted 15 December 2014 Available online xxxx Submitted by S. Kirkland MSC: 05C50 15A18 Keywords: Distance Laplacian eigenvalues Distance matrix Unicyclic graphs and trees
a b s t r a c t The distance Laplacian matrix L(G) of a graph G is defined to be L(G) = diag(Tr) − D(G), where D(G) denotes the distance matrix of G and diag(Tr) denotes the diagonal matrix of the vertex transmissions in G. Similarly, the distance signless Laplacian matrix of G is defined as Q(G) = diag(Tr) +D(G). The eigenvalues of L(G) and Q(G) are called the distance Laplacian and distance signless Laplacian eigenvalues, respectively. In this paper, four conjectures proposed by M. Aouchche and P. Hansen about the largest and the second largest distance Laplacian eigenvalues and the second largest distance signless Laplacian eigenvalue of a graph are proved. © 2014 Elsevier Inc. All rights reserved.
1. Introduction Throughout this paper, all graphs considered are simple, undirected and connected. A graph is denoted by G = G(V, E) with vertex set V (G) and edge set E(G). The * Corresponding author. 1
E-mail address: [email protected] (D. Wong). Supported by the National Natural Science Foundation of China (11171343).
http://dx.doi.org/10.1016/j.laa.2014.12.015 0024-3795/© 2014 Elsevier Inc. All rights reserved.
F. Tian et al. / Linear Algebra and its Applications 471 (2015) 10–20
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order of G is defined as the cardinality of the set V (G). The set of vertices adjacent to u ∈ V (G), denoted by N (u), refers to the neighborhood of u, and the degree of u means the cardinality of N (u). A pendent vertex is a vertex with degree one. The symbol u ∼ v means that two vertices u and v of G are adjacent. u and v are called twin points if NG (u) = NG (v) for u, v ∈ V (G). Generally, letting S ⊆ V (G), if NG (u) = NG (v) for any u, v ∈ S, we call S a set of twin points. If G is a tree, then a set of twin points consists of pendent vertices. The distance between u ∈ V (G) and v ∈ V (G), denoted by d(u, v), is the length of a shortest path between u and v. In particular, d(u, u) = 0 for any vertex u ∈ V (G). The diameter of G, denoted by d(G), is the maximum distance between vertices. The transmission Tr G (u) of a vertex u in G is defined as the sum of distances of u to all vertices of V (G), i.e., Tr G (u) = v∈V (G) d(v, u). The Wiener index W (G) of a graph G is defined as W (G) = 12 u,v∈V d(u, v). Obviously, W (G) = 12 v∈V Tr(v). Usually, we use Kn , Cn , Pn and Sn to denote the complete graph, the cycle, the path and the star with order n, respectively. By Sn+ we denote the graph obtained from Sn by adding an edge joining two pendent vertices of Sn . The complete bipartite graph with two color classes of order a and b is denoted by Ka,b . We denote by G − u, for u ∈ V (G), the induced subgraph of G obtained by deleting the vertex u and all the edges incident to it, and by G − e, for e ∈ E(G), we denote the subgraph of G obtained by deleting e from G. A subgraph H of G is called a spanning subgraph if V (H) = V (G), i.e., H is obtained from G by deleting some edges from G. Suppose that V (G) = {v1 , v2 , . . . , vn }. The distance matrix of G, denoted by D(G), is an n × n real symmetric matrix with the (i, j)-entry being d(vi , vj ). Obviously, the transmission Tr G (v) of a vertex v is the sum of all coordinates of the row vector of D(G) indexed by v. For a graph G, denote by diag(Tr) the diagonal matrix with the (i, i)-entry being Tr G (vi ). Distance Laplacian matrix introduced by M. Aouchiche and P. Hansen [1], is defined as L(G) = diag(Tr) −D(G). The eigenvalues of L(G), written as {∂1 (G), ∂2 (G), . . . , ∂n (G)}, are called the distance Laplacian eigenvalues of G. Sometimes we write the distance Laplacian eigenvalues of G as {∂1 , ∂2 , . . . , ∂n } for short, and labeled them such that ∂1 ≥ ∂2 ≥ . . . ≥ ∂n . Distance signless Laplacian matrix Q(G) is defined as Q(G) = diag(Tr) + D(G). Similarly, the distance eigenvalues and the distance signless Laplacian eigenvalues of a graph G of order n are respectively labeled as δ1 (G) ≥ δ2 (G) ≥ . . . ≥ δn (G); q1 (G) ≥ q2 (G) ≥ . . . ≥ qn (G). A real matrix A with nonnegative entries is called irreducible if for all i, j there is a k such that the (i, j)-entry of Ak is positive. For an n × n real symmetric matrix A, we denote by λ(A) one of its eigenvalues. Usually, the eigenvalues of A are labeled as λ1 (A) ≥ λ2 (A) ≥ . . . ≥ λn (A). Distance spectrum and distance signless Laplacian spectrum of graphs have attracted a lot of attention (see [3–10]). However, little is known for the distance Laplacian
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spectrum of graphs. In [1], M. Aouchiche and P. Hansen gave the distance Laplacian characteristic polynomials of some special graphs, and proved that the distance Laplacian eigenvalues do not decrease on deletion of edges. In [2], the authors introduced some properties for the distance Laplacian eigenvalues of graphs. Especially, they proposed nine conjectures on the largest and the second largest eigenvalue of the distance Laplacian matrix of a graph. This paper is devoted to prove three of the conjectures in [2] and a conjecture in [12], listed as follows. Conjectures (proposed by M. Aouchiche and P. Hansen in [2,12]). • Let G be a unicyclic graph with order n (≥ 6), then ∂1 (G) ≥ ∂1 (Sn+ ) with equality if and only if G = Sn+ . • For any graph G on n (≥ 4) vertices, ∂2 (G) ≥ n with equality if and only if G = Kn , or G = Kn − e for some e ∈ E(Kn ). • If T is a tree with order n (≥ 5), then ∂2 (T ) ≥ 2n − 1 with equality if and only if T = Sn . • Let T be a tree on n (≥ 4) vertices and q2 be the second largest eigenvalue of Q(T ). Then q2 ≥ 2n − 5, with equality if and only if T = Sn . 2. Preliminaries and lemmas Before giving the proof of the conjectures, we introduce some known conclusions used in the next section. At the end of this section, some plain however interesting results are given. Lemma 2.1 (Courant–Weyl inequality). (See [11, p. 31].) Let A1 and A2 be real symmetric matrices of order n. Then λn (A2 ) + λi (A1 ) ≤ λi (A1 + A2 )
for 1 ≤ i ≤ n.
Lemma 2.2 (Interlacing theorem). (See [11, p. 30].) Let A be a real symmetric matrix of order n and M be a principal submatrix of A with order s (≤ n). Then λi+n−s (A) ≤ λi (M ) ≤ λi (A),
1 ≤ i ≤ s.
Lemma 2.3 (Perron–Frobenius theorem). Let S = (sij ) be a complex matrix, and A be an irreducible matrix of the same order. |S| denotes the matrix whose (i, j)-entry is |sij |. If |S| ≤ A and S has σ as an eigenvalue, then |σ| ≤ λ1 (A). If the equality holds, then |S| = A, and there is a diagonal matrix E with diagonal entries of absolute value 1 and a constant c of absolute value 1, such that S = cEAE −1 .
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Lemma 2.4 (Gershgorin theorem). (See [2].) Let M = (mij ) be a complex n × n matrix and denote by λ1 , λ2 , . . . , λp its distinct eigenvalues. Then {λ1 , λ2 , . . . , λp } ⊂
n
z : |z − mii | ≤
i=1
|mij | .
j=i
Lemma 2.5. (See [2, Theorem 2.2].) For any graph G, ∂n (G) = 0 with multiplicity 1. Lemma 2.6. (See [1, Corollary 3.6].) Let G be a graph on n (≥ 3) vertices. Then qi (G) ≥ n − 2 for 2 ≤ i ≤ n. Lemma 2.7. (See [2, Theorem 2.3].) Let G be a graph of order n. Let S be a set of twin points of G, where S = {v1 , v2 , . . . , vk } and 1 < k < n. Then Tr(vi ) + 2 is an eigenvalue of L(G) with multiplicity at least k − 1. Lemma 2.8. (See [2, Theorem 2.7].) If G is a graph on n (≥ 2) vertices, then the multiplicity of the largest eigenvalue ∂1 of L(G) satisfies m(∂1 ) ≤ n − 1, with equality if and only if G = Kn . Lemma 2.9. (See [4, Lemma 2.3].) Let G be a unicyclic graph of order n (≥ 6), and G = Sn+ . Then W (G) ≥ n2 − n − 4 > W Sn+ . Lemma 2.10. (See [1, Theorem 3.5].) Let G be a graph on n vertices and with m (≥ n) ≥ ∂i (G) Then ∂i (G) edges. Denote the graph obtained by deleting an edge from G by G. for i = 1, 2, . . . , n. In the end of this section we give some plain results. For a graph G of order n, obviously, L(G) is a real symmetric matrix. Then the following proposition follows from Lemma 2.2 immediately. Proposition 2.11. Let G be a graph with order n, and let M be the principal submatrix of L(G) with order s ≤ n. Then ∂i+n−s ≤ λi (M ) ≤ ∂i ,
1 ≤ i ≤ s.
2
Let Tr max (G) be the maximal vertex transmission of a graph G. Then the following result can be easily derived from Lemma 2.2 and Lemma 2.4. Proposition 2.12. Tr max (G) ≤ ∂1 ≤ 2Tr max (G). Proof. Suppose v ∈ V (G) such that Tr G (v) = Tr max (G). For the left inequality, let A be a 1 × 1 matrix with the only entry being Tr G (v). Then A is a principal submatrix
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of L(G). Applying Lemma 2.2, we have Tr max (G) = λ1 (A) ≤ λ1 (L(G)) = ∂1 . The right inequality is a direct conclusion of Gershgorin theorem. 2 Proposition 2.13. Let G be a graph of order n (≥ 3). Then (a) 2δi (G) ≤ qi (G), 1 ≤ i ≤ n; (b) q1 (G) > ∂1 . Proof. Obviously, Q(G) = L(G) + 2D(G). Lemma 2.5 says that L(G) is positive semidefinite with 0 being the least eigenvalue, i.e., ∂n = 0. Applying Lemma 2.1, we have that qi (G) = λi (Q(G)) ≥ λi (2D(G)) + λn (L(G)) = 2δi (G) + 0 = 2δi (G). Obviously, |L(G)| = Q(G), and Q(G) is irreducible (see [4]). Applying Lemma 2.3, we have ∂1 = |∂1 | ≤ q1 (G). Here, the equality fails to hold, since the least eigenvalue of L(G) is 0 (using Lemma 2.5) and the least eigenvalue of Q(G) is at least n − 2 (using Lemma 2.6), forcing L(G) = cEQ(G)E −1 for any c with |c| = 1 and any invertible matrix E. 2 Remark 2.14. By the left inequality of Proposition 2.12 and by (b) of Proposition 2.13, we obtain a strict inequality q1 (G) > Tr max (G), which improves Lemma 2.3 in [3]. 3. Proof for the four conjectures In this section, we demonstrate the conjectures given at the end of Section 1. Theorem 3.1. Let G be a unicyclic graph with order n (≥ 6), then ∂1 (G) ≥ ∂1 (Sn+ ) with equality if and only if G = Sn+ . Proof. The explanation for Sn+ in [1] tells us the largest distance Laplacian eigenvalue of Sn+ with n > 3 is 2n − 1. Thus, to complete the proof we only need to prove that ∂1 (G) > ∂1 (Sn+ ) if G is a unicyclic graph of order n (≥ 6) which is different from Sn+ . If this is n the case, then by Lemma 2.9, W (G) ≥ n2 − n − 4 > W (Sn+ ). Since i=1 ∂i = 2W (G), it follows from Lemma 2.5 and Lemma 2.8 that (n − 1)∂1 > 2W (G) (also see [2]). Thus, we have ∂1 >
2(n2 − n − 4) 8 2W (G) ≥ = 2n − . n−1 n−1 n−1
Consequently, if n ≥ 9, then ∂1 > 2n −
8 ≥ 2n − 1. n−1
If 6 ≤ n ≤ 8, by a direct calculation we also obtain ∂1 > 2n − 1. Therefore, ∂1 (G) ≥ 2n − 1 = ∂1 (Sn+ ), and the equality holds if and only if G = Sn+ . 2
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Theorem 3.2. Let G be a graph on n (≥ 4) vertices. Then ∂2 (G) ≥ n, with equality if and only if G = Kn or G = Kn − e with e ∈ E(Kn ).
n be the spanning subgraph of Kn obtained by deleting two different edges, Proof. Let K
n ) > n. say e1 , e2 , from Kn . Firstly, we prove that ∂2 (K Case 1. e1 and e2 have a common vertex. Suppose the common vertex of e1 and e2 is v1 and the other two end vertices are v2
n ), written as M , indexed by v1 , v2 and v3 . Considering the principal submatrix of L(K and v3 , we have ⎛
n + 1 −2 ⎝ M= −2 n −2 −1
⎛ ⎞ −2 1 ⎝ ⎠ −1 = nI3 + −2 n −2
⎞ −2 −2 0 −1 ⎠ . −1 0
The second largest eigenvalue of M is λ2 (M ) = n + 1. By Proposition 2.11, we have
n ) ≥ λ2 (M ) = n + 1 > n. ∂2 (K Case 2. e1 and e2 have no common vertices. Let the end vertices of e1 and e2 be v1 , v2 and v3 , v4 , respectively. Let the principal
n indexed by v1 , v2 , v3 , v4 be M1 . Then, submatrix of K ⎛
n −2 −1 ⎜ −2 n −1 M1 = ⎜ ⎝ −1 −1 n −1 −1 −2
⎛ ⎞ −1 0 −2 −1 ⎜ −2 0 −1 −1 ⎟ ⎟ = nI4 + ⎜ ⎝ −1 −1 0 −2 ⎠ n −1 −1 −2
⎞ −1 −1 ⎟ ⎟. −2 ⎠ 0
The second largest eigenvalue of M1 is λ2 (M1 ) = n + 2. Applying Proposition 2.11 again, we have
n ) ≥ λ2 (M1 ) = n + 2 > n. ∂2 (K Let G be a graph of order n. If G = Kn and G = Kn − e, then it can be viewed
n by deleting some edges. Applying Lemma 2.10, we have as a graph obtained from K
n ) > n. On the other hand, if G = Kn or G = Kn − e, from 1.1, 1.2 in ∂2 (G) ≥ ∂2 (K [1] it follows that ∂2 (G) = n. Finally, we have that ∂2 (G) ≥ n, and the equality holds if and only if G = Kn or G = Kn − e with e ∈ E(Kn ). 2 The following two lemmas will be applied in the proof of the third conjecture. Let T be a tree with diameter d. Suppose P = v1 v2 . . . vd+1 is a longest path in T such that vi ∼ vi+1 for i = 1, 2, . . . , d, then P is a diameter of tree T . For each vi ∈ V (P ), denote by Pvi the induced connected subgraph of T with maximum possible vertices, which contains the vertex vi and contains no other vertices of P . Now we construct a
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Fig. 1. The example of T and T .
new tree (with the same order and diameter as T ) from T in the following way: for each vi ∈ V (P ), we delete all the edges in Pvi and add edges joining vi . The resultant tree is denoted by T (see Fig. 1 for example). Lemma 3.3. Let T , T and P be as described above. Then max Tr T (v1 ), Tr T (vd+1 ) ≥ max Tr T (v1 ), Tr T (vd+1 ) , where d = d(T ), and v1 , vd+1 are the end vertices of P . / V (P ), then Proof. For v ∈ V (T ), if v ∈ V (P ), then dT (v1 , v) = dT (v1 , v); if v ∈ dT (v1 , v) ≥ dT (v1 , v). Consequently,
Tr T (v1 ) =
dT (v1 , v) ≥
v∈V (T )
v∈V
dT (v1 , v) = Tr T (v1 ).
(T )
Similarly, Tr T (vd+1 ) ≥ Tr T (vd+1 ). This completes the proof. 2 Lemma 3.4. Let T be a tree with diameter d. Then Tr max (T ) ≥
1 n(d + 2) − (d + 1). 2
Proof. Let P = v1 v2 . . . vd+1 and T be defined as above. Clearly, in T , for a vertex v outside P , d(v1 , v) + d(vd+1 , v) = d + 2, and for a vertex v of P , d(v1 , v) + d(vd+1 , v) = d. Thus, we have Tr T (v1 ) + Tr T (vd+1 ) = (n − d − 1)(d + 2) + (d + 1)d = n(d + 2) − 2(d + 1), from which it follows that
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1 max Tr T (v1 ), Tr T (vd+1 ) ≥ Tr T (v1 ) + Tr T (vd+1 ) 2 1 = n(d + 2) − (d + 1). 2 Applying Lemma 3.3, we have Tr max (T ) ≥ max Tr T (v1 ), Tr T (vd+1 ) ≥ max Tr T (v1 ), Tr T (vd+1 ) ≥
1 n(d + 2) − (d + 1). 2
2
Remark 3.5. By the proof of Lemma 3.4, one can easily see that max{Tr T (v1 ), Tr T (vd+1 )} ≥ 12 n(d + 2) − (d + 1), where v1 , vd+1 are the end vertices of a longest path and d = d(T ). Now we are in a position to prove the third conjecture. Theorem 3.6. Let T be a tree with order n (≥ 5). The second largest Laplacian eigenvalue ∂2 (T ) of T satisfies ∂2 (T ) ≥ 2n − 1, and the equality holds if and only if T = Sn . Proof. Let T be a tree with diameter d. Suppose P = v1 v2 . . . vd+1 is a longest path in G such that vi ∼ vi+1 for i = 1, 2, . . . , d. Suppose the set of twin points containing v1 (resp., vd+1 ) is W1 (resp., Wd+1 ). Let n1 = |W1 | and nd+1 = |Wd+1 |. The description of 1.3 in [1] tells us the second largest distance Laplacian eigenvalue of Sn is 2n − 1. Thus, to complete the proof we only need to prove that ∂2 (T ) > 2n − 1 if T = Sn and T has at least 5 vertices. If T = Sn , then d ≥ 3. We complete the proof by considering three different cases for d. Case 1. d = 3. Assume n1 or nd+1 is more than 2. Suppose n1 > 2, without loss of generality. Then, it follows from Lemma 2.7 that T has Tr T (v1 ) + 2 as its distance Laplacian eigenvalue with multiplicity at least n1 − 1 (≥ 2). Obviously, Tr T (v1 ) = d(v1 , v1 ) + d(v1 , v2 ) + d(v1 , v4 ) + ≥0+1+3+
d(v, v1 )
v ∈{v / 1 ,v2 ,v4 }
2 = 4 + 2(n − 3) = 2n − 2.
v ∈{v / 1 ,v2 ,v4 }
Thus we have ∂2 (T ) ≥ Tr T (v1 ) + 2 ≥ 2n > 2n − 1. If n1 = 2 and nd+1 = 2, from Lemma 2.7, T has Tr T (v1 ) + 2 and Tr T (vd+1 ) + 2 as its distance Laplacian eigenvalues with multiplicity at least n1 − 1 (= 1) and at least nd+1 − 1 (= 1), respectively. Similar as the case that n1 > 2, Tr T (v1 ) ≥ 2n − 2 and Tr T (vd+1 ) ≥ 2n − 2, from which we have ∂2 (T ) ≥ 2n > 2n − 1.
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Fig. 2. The graphs G1 , G2 , . . . , G8 .
For the left subcases of d = 3, T has five vertices (see G1 in Fig. 2). By a direct calculation we confirm that ∂2 (T ) > 2n − 1. Case 2. d = 4. If n1 > 2, or nd+1 > 2, or n1 = nd+1 = 2, by similar discussions as Case 1, we have ∂2 (T ) > 2n − 1. The parallel discussions are omitted. If n1 + nd+1 = 3, say n1 = 2, nd+1 = 1, without loss of generality. Assume the principal submatrix of L(T ) indexed by Tr T (v1 ) −2 v1 and its twin point, say v1 , is M0 = . It is easy to see that λ2 (M0 ) = −2 Tr T (v1 ) Tr T (v1 ) − 2. Clearly, 5 Tr T (v1 ) = d v1 , v1 + d(v1 , vi ) + i=1
d(v1 , v)
v ∈{v / i ,v1 :1≤i≤5,}
≥ 2 + 10 + 3(n − 6) = 3n − 6. Consequently, we have λ2 (M0 ) = Tr T (v1 ) − 2 ≥ 3n − 8. If n > 7, then λ2 (M0 ) > 2n − 1. So, ∂2 (T ) ≥ λ2 (M0 ) > 2n − 1. When n = 6 or n = 7 in this subcase, we confirm the conclusion by a direct calculation. In the remainder of this case we may assume that each end vertex of every longest path has no other twin points other than itself (see G2 in Fig. 2). Let M be the principal Tr T (v1 ) −4 submatrix indexed by v1 , v5 of G2 . Then M = . Obviously, Tr T (v1 ) = −4 Tr T (v5 ) Tr T (v5 ), and Tr T (v1 ) = d(v1 , v1 ) + d(v1 , v2 ) + d(v1 , v3 ) + d(v1 , v5 ) + ≥0+1+2+4+
d(v, v1 )
v ∈{v / 1 ,v2 ,v3 ,v5 }
3 = 7 + 3(n − 4) = 3n − 5.
v ∈{v / 1 ,v2 ,v3 ,v5 }
Applying Proposition 2.11, we have ∂2 (T ) ≥ λ2 (M ) = Tr T (v1 ) − 4, from which it follows that ∂2 (T ) ≥ Tr T (v1 ) − 4 ≥ 3n − 9. If n ≥ 9, then ∂2 (T ) > 2n − 1. If 5 ≤ n ≤ 8 (see G3 , . . . , G8 in Fig. 2), the conclusion is verified by a direct calculation.
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Case 3. d ≥ 5. Without loss of generality, we suppose that Tr T (v1 ) ≥ Tr T (vd+1 ). As v2 is adjacent to v1 , Tr T (v1 ) = Tr T (v2 ) + n − 2. Let M1 be the principal submatrix of L(T ) indexed by v1 and v2 . Then M1 =
−1 Tr T (v1 ) −1 Tr T (v2 )
= Tr T (v1 )I2 +
0 −1 −1 2 − n
.
The second largest eigenvalue of M1 is λ2 (M1 ) = Tr T (v1 ) − Remark 3.5 tells us that Tr T (v1 ) ≥ (n − 2)2 + 4 ≤ n. Hence,
1 (n − 2) + (n − 2)2 + 4 . 2 1 2 n(d
+ 2) − (d + 1), and it is obvious that
1 n(d + 2) − (d + 1) − (n − 1) λ2 (M1 ) ≥ 2
=
1 nd − d. 2
Applying Proposition 2.11, we obtain ∂2 (T ) ≥ λ2 (M1 ) ≥ 12 nd − d. If n > 8, it is clear that ∂2 (T ) ≥ 12 nd − d > 2n − 1 for d ≥ 5. If 6 ≤ n ≤ 8, we confirm ∂2 (T ) > 2n − 1 by a direct calculation. Finally, we have ∂2 (T ) ≥ ∂2 (Sn ) = 2n − 1, and the equality holds if and only if T = Sn . 2 Ultimately, we prove the last conjecture first proposed in [12]. Theorem 3.7. Let T be a tree on n (≥ 4) vertices and q2 be the second largest eigenvalue of Q(T ). Then q2 ≥ 2n − 5, with equality if and only if T = Sn . Proof. As in Theorem 3.6, let the diameter of T be d. Suppose P = v1 v2 . . . vd+1 is a longest path in G such that vi ∼ vi+1 for i = 1, 2, . . . , d. Suppose the set of twin points containing v1 (resp., vd+1 ) is W1 (resp., Wd+1 ). Let n1 = |W1 | and nd+1 = |Wd+1 |. Section 1.3 in [1] tells us that q2 (Sn ) = 2n − 5 if n ≥ 4. To complete the proof it suffices to prove that q2 (T ) > 2n − 5 if T = Sn and n ≥ 4. Case 1. d = 3. If n = 4, then T = P = P4 . By a direct calculation, we obtain the result. If n > 4, then one of the end vertices of P , say v1 , has a twin point, say v1 , which is different from v1 . Therefore, Tr T (v1 ) = Tr T v1 ≥ 2(n − 4) + 1 + 2 + 3 = 2n − 2.
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Tr T (v1 ) 2 Let the principal submatrix of Q(T ), indexed by v1 and v1 , be M = . 2 Tr T (v1 ) The second largest eigenvalue of M is λ2 (M ) = Tr T (v1 ) − 2. Thus, by Proposition 2.11, we have q2 ≥ λ2 (M ) = Tr T (v1 ) − 2 ≥ 2n − 4 > 2n − 5. Case 2. d ≥ 4. Similar as Case 3 of Theorem 3.6, the principal submatrix of Q(T ) indexed by v1 and v2 , is M1 =
Tr T (v1 ) 1
1 Tr T (v2 )
= Tr T (v1 )I2 +
0 1 1 2−n
,
since Tr T (v1 ) = Tr T (v2 ) + n − 2. For the second largest eigenvalue of M1 , 1 (n − 2) + (n − 2)2 + 4 2 1 ≥ Tr T (v1 ) − n + 1 ≥ nd − d. 2
λ2 (M1 ) = Tr T (v1 ) −
So, we have q2 ≥ λ2 (M1 ) ≥ 12 nd − d. If n > 4, then q2 ≥ 12 nd − d > 2n − 5, as desired. Finally, we have that q2 > 2n − 5 if n ≥ 4, and the equality holds if and only if T = Sn . 2 References [1] M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013) 21–33. [2] M. Aouchiche, P. Hansen, Some properties of the distance Laplacian eigenvalues of a graph, Les Cahiers du GERAD, G-2013-28, http://www.gerad.ca/fichiers/cahiers/G-2013-28.pdf. [3] Rundan Xing, Bo Zhou, On the distance and distance signless Laplacian spectral radii of bicyclic graphs, Linear Algebra Appl. 439 (2013) 3955–3963. [4] R. Xing, B. Zhou, J. Li, On the distance signless Laplacian spectral radius of graphs, Linear Multilinear Algebra 62 (2014) 1377–1387. [5] A. Ilić, Distance spectral radius of trees with given matching number, Discrete Appl. Math. 158 (2010) 1799–1806. [6] M. Nath, S. Paul, On the distance spectral radius of bipartite graphs, Linear Algebra Appl. 436 (2012) 1285–1296. [7] D. Stevanović, A. Ilić, Distance spectral radius of trees with fixed maximum degree, Electron. J. Linear Algebra 20 (2010) 168–179. [8] S.S. Bose, M. Nath, S. Paul, On the maximal distance spectral radius of graphs without a pendent vertex, Linear Algebra Appl. 438 (2013) 4260–4278. [9] S.N. Ruzieh, D.L. Powers, The distance spectrum of the path Pn and the first distance eigenvector of connected graphs, Linear Multilinear Algebra 28 (1990) 75–81. [10] G. Yu, On the least distance eigenvalue of a graph, Linear Algebra Appl. 439 (8) (2013) 2428–2433. [11] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, New York, 2012. [12] M. Aouchiche, P. Hansen, A signless Laplacian for the distance matrix of a graph, Les Cahiers du GERAD, G-2011-78, http://www.gerad.ca/fichiers/cahiers/G-2011-78.pdf.