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Further results on the spectral radius of matrices and graphs
Applied Mathematics and Computation 239 (2014) 326–332
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Further results on the spectral radius of matrices and graphs q Wenxi Hong, Lihua You ⇑ School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China
a r t i c l e
i n f o
Keywords: Distance Signless Laplacian Spectral radius Bounds Graph
a b s t r a c t In this paper, we obtain some results on the sharp upper bounds of the spectral radius of a nonnegative matrix, then apply these results to signless Laplacian matrices to obtain some known results. We also apply these results to distance signless Laplacian matrix of a graph, and obtain some sharp bounds on the distance signless Laplacian spectral radius. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Let A ¼ ðaij Þ be an n n nonnegative matrix. The spectral radius of A, denoted by kðAÞ, is the largest modulus of eigenvalues of A. Furthermore, if A is irreducible, then kðAÞ is just the largest eigenvalue of A. Let G be a simple graph with vertex set VðGÞ ¼ fv 1 ; v 2 ; . . . ; v n g. Let AðGÞ ¼ ðaij Þnn be the ð0; 1Þ–adjacency matrix of G, where aij ¼ 1 if v i and v j are adjacent and 0 otherwise. Let di be the degree of the vertex v i in G for i ¼ 1; 2; . . . ; n and satisfy d1 P d2 P P dn and DðGÞ ¼ diagðd1 ; d2 ; . . . ; dn Þ be the degree diagonal matrix. Then Q ðGÞ ¼ DðGÞ þ AðGÞ is the signless Laplacian matrix of G. Clearly, Q ðGÞ is symmetric. Let q1 ðGÞ be the spectral radius of Q ðGÞ and it is also called the signless Laplacian spectral radius of G. The distance matrix DðGÞ ¼ ðdij Þ of G, its ði; jÞ–entry, dij , is equal to dG ðv i ; v j Þ which denotes the distance (the length of the shortest path) between vertex v i and vertex v j . Clearly, the distance matrix of a connected graph is irreducible and symmetric. The eigenvalues of DðGÞ are given as d1 ðGÞ P d2 ðGÞ P P dn ðGÞ, where d1 is called the distance spectral radius of G. The transmission of v i is defined to be the sum of the distance from v i to all other vertices in G, denoted by Di , that is, P Di ¼ nj¼1 dij . The transmission is also called the first distance degree [5]. Assume that the transmissions are ordered as D1 P D2 P P Dn . A connected graph G is said to be k–transmission regular if Di ¼ k for all i 2 f1; 2; . . . ; ng. Let TrðGÞ ¼ diagðD1 ; D2 ; . . . ; Dn Þ denote the diagonal matrix of its vertex transmissions. Aouchiche and Hansen [1] introduced the Laplacian and the signless Laplacian for the distance matrix of a connected graph. The matrix DL ðGÞ ¼ TrðGÞ DðGÞ is called the distance Laplacian of G, while the matrix DQ ðGÞ ¼ TrðGÞ þ DðGÞ is called the distance signless Laplacian of G. Let G be a connected graph, then the matrix DQ ðGÞ ¼ ðqij Þ is symmetric, nonnegative and irreducible. All the eigenvalues of DQ ðGÞ can be arranged as: dQ1 ðGÞ P dQ2 ðGÞ P P dQn ðGÞ. dQ1 ðGÞ is called the distance signless Laplacian spectral radius of G. As the DQ ðGÞ is irreducible, by the Perron–Frobenius theorem, dQ1 ðGÞ is positive, simple and there is a unique positive unit eigenvector x corresponding to dQ1 ðGÞ, which is called the distance signless Perron vector of G.
q Research supported by the Zhujiang Technology New Star Foundation of Guangzhou (No. 2011J2200090), and Program on International Cooperation and Innovation, Department of Education, Guangdong Province (No. 2012gjhz0007). ⇑ Corresponding author. E-mail address: [email protected] (L. You).
http://dx.doi.org/10.1016/j.amc.2014.04.107 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
327
As usual, we denote by K n the complete graph, by C n the cycle and by K a;na the complete bipartite graph. In [1], Aouchiche and Hansen obtain the lower sharp bound as dQ1 ðGÞ P dQ1 ðK n Þ ¼ 2n 2. In Section 2, we obtain some results on the sharp upper bounds of the spectral radius of a nonnegative matrix, and apply these results to the signless Laplacian matrix to obtain some known results. In Section 3, we present some sharp bounds on the distance signless Laplacian spectral radius of graphs. 2. Further results on the spectral radius of nonnegative matrices In this section, we obtain some new results on the sharp upper bounds of the spectral radius of nonnegative matrices. Then we apply these results to the signless Laplacian matrix and obtain some known results. Lemma 2.1 [7]. If A is an n n nonnegative matrix with the spectral radius kðAÞ and row sums r 1 ; r 2 ; . . . ; rn , then min16i6n r i 6 kðAÞ 6 max16i6n r i . Moreover, if A is irreducible, then one of the equalities holds if and only if the row sums of A are all equal. By Lemma 2.1 and the Cauchy–Weil inequality, we can get the following proposition. Proposition 2.2. Let A be an n n nonnegative, irreducible and symmetric matrix with row sums r 1 ; r2 ; . . . ; r n , where r 1 P r 2 P . . . P r n ; B ¼ A þ M where M ¼ diagðr1 ; r2 ; . . . ; r n Þ, and kðAÞ ðkðBÞÞ the spectral radius of A ðBÞ. Then kðBÞ 6 kðAÞ þ r 1 with equality if and only if the row sums of A are all equal. Let G be a simple and connected graph. Let A be the adjacency matrix of G and M be the degree diagonal matrix of G. Then B ¼ A þ M is the signless Laplacian matrix of G. So we can get the following corollary by Proposition 2.2. Corollary 2.3 [4, Lemma 9]. Let G be a simple and connected graph on n vertices, k1 ðGÞ be the spectral radius of G; q1 ðGÞ be the signless Laplacian spectral radius of G and D be the maximum degree of G. Then q1 ðGÞ 6 k1 ðGÞ þ D with equality if and only if G is regular. Theorem 2.4. Let A ¼ ðaij Þ be an n n nonnegative and irreducible matrix with aii ¼ 0 for i ¼ 1; 2; . . . ; n and row sums P r 1 ; r2 ; . . . ; r n with r 1 P r2 P P r n . Let B ¼ A þ M, where M ¼ diagðr1 ; r2 ; . . . ; r n Þ, si ¼ nj¼1 aij r j for any i 2 f1; 2; . . . ; ng and kðBÞ be the spectral radius of B. Then
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8
ð2:1Þ
with equality if and only if r 1 þ rs11 ¼ r 2 þ rs22 ¼ ¼ rn þ rsnn .
r i ; if i ¼ j; then B and M 1 BM are nonnegative and aij ; if i – j; 1 irreducible, and M BM and B have the same eigenvalue. Let x ¼ ðx1 ; x2 ; . . . ; xn ÞT be a positive eigenvector of M 1 BM corresponding to kðBÞ. We assume that one entry, say xp , is equal to 1 and the others are less than or equal to 1, that is, xp ¼ 1 and 0 < xk 1 for any k. Let xq ¼ maxfxk j1 6 k 6 n and k – pg. From M 1 BMx ¼ kðBÞx, we have Proof. Since A is irreducible, r i > 0, and B ¼ ðbij Þ, where bij ¼
kðBÞ ¼ kðBÞxp ¼
n X bpk r k xk k¼1
rp
¼ rp þ
n n X X bpk r k xk apk r k xk ¼ rp þ ; rp rp k¼1;k–p k¼1;k–p
thus
kðBÞ ¼ r p þ
n n X 1X 1 2 apk r k xk 6 r p þ xq apk r k rp k¼1 rp k¼1
! ¼ rp þ
xq s p ; rp
ð2:2Þ
with equality if and only if xk ¼ xq for all k 2 f1; 2; . . . ; ng n fpg, and
kðBÞxq ¼
n X bqk r k xk k¼1
rq
¼ r q xq þ
n n 1X 1X sq aqk r k xk 6 rq xq þ aqk r k ¼ r q xq þ ; r q k¼1 r q k¼1 rq
ð2:3Þ
with equality if and only if xk ¼ xp ¼ 1 for all k 2 f1; 2; . . . ; ng n fqg. From (2.2) and (2.3), we get n 1X xq s p apk r k xk ¼ kðBÞ rp 6 ; rp k¼1 rp
ð2:4Þ
n 1X sq aqk r k xk ¼ ðkðBÞ r q Þxq 6 : rq k¼1 rq
ð2:5Þ
and
328
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
From (2.4) and (2.5), we get kðBÞ r p > 0 and kðBÞ r q > 0 easily, and
k2 ðBÞ ðr p þ r q ÞkðBÞ þ r p r q
sp sq 6 0: rp rq
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Thus we have kðBÞ 6
r p þr q þ
4sp sq q
ðr p r q Þ2 þ rp r
i j
, then (2.1) holds. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8
6 max16i;j6n
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8
:
;
() x1 ¼ x2 ¼ ¼ xn ¼ 1 () ðM 1 BMÞð1; 1; . . . ; 1ÞT ¼ kðBÞð1; 1; . . . ; 1ÞT () BðMð1; 1; . . . ; 1ÞT Þ ¼ kðBÞðMð1; 1; . . . ; 1ÞT Þ () Bðr1 ; r2 ; . . . ; rn ÞT ¼ kðBÞðr 1 ; r2 ; . . . ; r n ÞT P P () r 2i þ si ¼ r 2i þ nj¼1 aij rj ¼ r2i þ nj¼1;j–i aij r j ¼ kðBÞri for any i 2 f1; 2; . . . ; ng () kðBÞ ¼ r i þ rsi for any i 2 f1; 2; . . . ; ng: i
) r 1 þ sr11 ¼ r2 þ rs22 ¼ ¼ r n þ rsnn : Now we show if r 1 þ rs11 ¼ r 2 þ rs22 ¼ ¼ r n þ rsnn , then kðBÞ ¼ r i þ rsii for any i 2 f1; 2; . . . ; ng. Let k ¼ r1 þ rs11 ¼ r 2 þ rs22 ¼ ¼ r n þ rsnn , then T
Bðr 1 ; r 2 ; . . . ; r n ÞT ¼ ðr 21 þ s1 ; r 22 þ s2 ; . . . ; r 2n þ sn Þ ¼ kðr 1 ; r 2 ; . . . ; r n ÞT ; it implies that k is an eigenvalue of B, so k 6 kðBÞ. 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9
P
v k v i dk be the average degree of the neighbors of
v i , where vertices v i and v k
are adjacent in G.
Lemma 2.5 [3]. Let G be a connected graph. For v i 2 VðGÞ, the degree of v i and the average degree of the vertices adjacent to v i are denoted by di and mi , respectively. Then d1 þ m1 ¼ d2 þ m2 ¼ ¼ dn þ mn holds if and only if G is a regular graph or a bipartite semiregular graph. We apply Theorem 2.4 to the signless Laplacian matrix Q ðGÞ ¼ DðGÞ þ AðGÞ of Q, then we can obtain the following corollary easily by Lemma 2.5. Corollary 2.6 [6, Theorem 6(b)]. Let G be a simple and connected graph on n vertices with degrees sequence fd1 ; d2 ; . . . ; dn g P which satisfies d1 P d2 P P dn and q1 ðGÞ be the signless Laplacian spectral radius of G. Let si ¼ v k v i dk , where vertices v i and v k are adjacent, for i 2 f1; 2; . . . ; ng. Then
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 i sj
bors of the vertex
i
v i . Let AðDÞ ¼ ðaij Þ denote the adjacency matrix of D, where aij
is equal to the number of arc ðv i ; v j Þ. We þ
þ
þ
apply Theorem 2.4 to the signless Laplacian matrix Q ðDÞ of a digraph D, where Q ðDÞ ¼ diagðd1 ; d2 ; . . . ; dn Þ þ AðDÞ. Then we can obtain the following corollary easily. Corollary 2.7 [2, Theorem 3.2]. Let D be a strongly connected digraph on n vertices. Then
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 2 > > þ þ þ þ > > þ 4mþi mþj di dj > 2 > > : ; Remark 2.8. By Theorem 2.4, we þ þ þ þ þ d1 þ mþ 1 ¼ d2 þ m2 ¼ ¼ dn þ mn .
conclude
that
the
equality
in
Corollary
2.7
holds
if
and
only
if
329
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
Theorem 2.9. Let A ¼ ðaij Þ be an n n nonnegative matrix with row sums r1 ; r2 ; . . . ; r n , where r 1 P r2 P P r n ; M be the largest diagonal element, N be the largest non-diagonal element and kðAÞ be the spectral radius of A. Then for i ¼ 1; 2; . . . ; n,
kðAÞ 6
ri þ M N þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðri þ N MÞ2 þ 4ði 1Þðr1 ri ÞN 2
ð2:6Þ
:
Moreover, if A is irreducible, when i ¼ 1, then the equality holds if and only if the row sums of A are all equal; when 2 6 i 6 n, then the equality holds if and only if A satisfies the following conditions: (i) all ¼ M for 1 6 l 6 i 1; (ii) alk ¼ N for 1 6 l; k 6 n and l – k; (iii) r1 ¼ r2 ¼ ¼ r i1 P r i ¼ r iþ1 ¼ ¼ r n . Proof. When i ¼ 1 or r 1 ¼ r i , it is clearly that the inequality is true and the equality holds if and only if the row sums of A are all equal by Lemma 2.1. We suppose r1 > r i and 2 6 i 6 n. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 x; 1 6 j 6 i 1; i þNMÞ þ4ði1Þðr 1 r i ÞN Let x ¼ Mri þð2i3ÞNþ ðr2ði1ÞN . It is easy to show that x > 1. Let xj ¼ Let 1; i 6 j 6 n: 1 U ¼ diagðx1 ; x2 ; . . . ; xn Þ be a diagonal matrix of order n. Let B ¼ U AU and note that B and A have the same eigenvalues, then kðBÞ ¼ kðAÞ. Now we consider the row sums s1 ; s2 ; . . . ; sn of matrix B. For 1 6 l 6 i 1, we have
sl ¼
n X xj j¼1
xl
alj ¼
i1 i1 i1 n n X 1X 1X 1 X 1 1 X alj þ alj ¼ alj þ 1 alj ¼ r l þ 1 alj ; x j¼i x j¼1 x j¼1 x x j¼1 j¼1
and for i 6 l 6 n,
sl ¼
n X xj j¼1
xl
alj ¼
n i1 i1 X X X alj þ ðx 1Þ alj ¼ rl þ ðx 1Þ alj : j¼1
j¼1
j¼1
Since A is a nonnegative matrix, then 0 6 apq 6 N if p – q, where 1 6 p; q 6 n. Thus 1 6 l 6 i 1, and
Pi1
j¼1 alj
6 M þ ði 2ÞN, where
i1 X alj 6 ði 1ÞN; where i 6 l 6 n: j¼1
As x > 1 and r1 P r 2 P P rn , then for 1 6 l 6 i 1,
sl 6
1 1 1 1 1 1 ½M þ ði 2ÞN ¼ ðrl MÞ þ M þ 1 ði 2ÞN 6 ðr 1 MÞ þ M þ 1 ði 2ÞN; rl þ 1 x x x x x x
with equality if and only if ðaÞ and ðbÞ hold: ðaÞall ¼ M and alj ¼ N if 1 6 j 6 i 1 with j – i; ðbÞr 1 ¼ rl . For i 6 l 6 n, we have sl 6 rl þ ðx 1Þði 1ÞN 6 r i þ ðx 1Þði 1ÞN, with equality if and only if ðcÞ and ðdÞ hold: ðcÞalj ¼ N if 1 6 j 6 i 1; ðdÞri ¼ rl . Noting that
M þ ri N þ 1 1 ði 2ÞN ¼ ri þ ðx 1Þði 1ÞN ¼ ðr1 MÞ þ M þ 1 x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
Mþr Nþ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr i þ N MÞ2 þ 4ði 1Þðr 1 r i ÞN 2
;
ðr þNMÞ þ4ði1Þðr r ÞN
1 i i i thus kðAÞ ¼ kðBÞ 6 maxfs1 ; s2 ; . . . ; sn g 6 . 2 When equality holds in (2.6), then ðaÞ ðbÞ hold for 1 6 l 6 i 1; ðcÞ ðdÞ hold for i 6 l 6 n. Thus all ¼ M for 1 6 l 6 i 1, r 1 ¼ r 2 ¼ ¼ r i1 > ri ¼ r iþ1 ¼ ¼ r n and all the non-diagonal elements are equal. Now (i)–(iii) follow. Conversely, if (i)–(iii) hold, it is easy to show that equality holds. h
Similarly, we apply Theorem 2.9 to Q ðGÞ ¼ DðGÞ þ AðGÞ and note that r i ¼ 2di for any i 2 f1; 2; . . . ; ng; M ¼ d1 and N ¼ 1. Then we have Corollary 2.10 [9, Theorem 3.2]. Let G be a simple and connected graph with n vertices and degrees sequence fd1 ; d2 ; . . . ; dn g which satisfies d1 P d2 P P dn and q1 ðGÞ be the signless Laplacian spectral radius of G. Then for i ¼ 1; 2; . . . ; n,
q1 ðGÞ 6
2di þ d1 1 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2di þ 1 d1 Þ þ 8ði 1Þðd1 di Þ 2
:
330
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
Moreover, if i ¼ 1, then the equality holds if and only if G is a regular graph; if 2 6 i 6 n, then the equality holds if and only if G is either a regular graph or a bidegreed graph in which d1 ¼ d2 ¼ ¼ di1 ¼ n 1 and di ¼ diþ1 ¼ ¼ dn . 3. Bounds on distance signless Laplacian spectral radius In this section, we present some sharp bounds on the distance signless Laplacian spectral radius by the above theorems and lemmas in Section 2. P Noting that the ith row sum of DðGÞ (DQ ðGÞ) is ri ¼ Di (ri ¼ Di þ nj¼1 dij ¼ 2Di ), we can obtain the following results by Lemma 2.1 easily. Proposition 3.1. Let G be a simple connected graph on n vertices, D1 and Dn be the largest and least transmissions of G, respectively. Then Dn 6 d1 ðGÞ 6 D1 . Moreover, one of the equalities holds if and only if G is a transmission regular graph.
Proposition 3.2. Let G be a simple connected graph on n vertices, D1 and Dn be the largest and least transmissions of G, respectively. Then 2Dn 6 dQ1 ðGÞ 6 2D1 . Moreover, one of the equalities holds if and only if G is a transmission regular graph. By Proposition 3.2, we obtain the largest distance signless Laplacian spectral radius of K n ; K n2;n2 and C n as follows.
dQ1 ðK n Þ
¼ 2ðn 1Þ;
dQ1
K 2n;2n ¼ 3n 4 and dQ1 ðC n Þ ¼
(
n2 1 ; 2 n2 2
;
if n is odd; if n is even:
Lemma 3.3 [1]. Let G be a connected graph on nðP 2Þ vertices. Then dQ1 ðGÞ P dQ1 ðK n Þ ¼ 2ðn 1Þ and dQi ðGÞ P dQi ðK n Þ ¼ n 2, for all 2 6 i 6 n. Moreover, dQ2 ðGÞ ¼ dQ2 ðK n Þ ¼ n 2 if and only if G is the complete graph K n .
Lemma 3.4 [8]. Let A be a positive semidefinite Hermitian matrix, k1 ; k2 ; . . . ; kn be the eigenvalues of A where k1 P k2 P P kn and a1 ; a2 ; . . . ; an be the entries of the main diagonal of A satisfying a1 P a2 P P an . Then the spectrum of A majorizes its main P P P P diagonal, that is, ki¼1 ki P ki¼1 ai for k ¼ 1; 2; . . . ; n, and ni¼1 ki ¼ ni¼1 ai . Theorem 3.5. Let G be a simple connected graph on n vertices, d1 ðGÞ be the distance spectral radius of G and D1 be the largest transmission of G. Then D1 6 dQ1 ðGÞ 6 d1 ðGÞ þ D1 . Moreover, dQ1 ðGÞ ¼ d1 ðGÞ þ D1 if and only if G is a transmission regular graph. Proof. Denote the spectrum of DQ ðGÞ by spðDQ ðGÞÞ ¼ ðdQ1 ðGÞ; dQ2 ðGÞ; . . . ; dQn ðGÞÞ and the transmissions D1 ; D2 ; . . . ; Dn of G satisfy D1 P D2 P P Dn . Noting that dQn ðGÞ P dQn ðK n Þ ¼ n 2 P 0 for n P 2 by Lemma 3.3, then DQ ðGÞ is a positive semidefinite Hermitian matrix. By Lemma 3.4, we have dQ1 ðGÞ P D1 . By Proposition 2.2, we directly get the dQ1 ðGÞ 6 d1 ðGÞ þ D1 with equality if and only if G is a transmission regular graph. h
Definition 3.6 [5]. Let G be a simple connected graph on n vertices, DðGÞ ¼ ðdij Þ be the distance matrix and the transmission sequence fD1 ; D2 ; . . . ; Dn g with D1 P D2 P P Dn . Then the second distance degree of v i , denoted by T i , is given by P T i ¼ ni¼1 dij Di . We can easily obtain Theorems 3.7 and 3.8 from Theorems 2.4 and 2.9, respectively. Theorem 3.7. Let G be a simple connected graph on n vertices, fD1 ; D2 ; ; Dn g be the transmission sequence of G with D1 P D2 P P Dn and fT 1 ; T 2 ; . . . ; T n g be the second distance degree sequence of G. Then
dQ1 ðGÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 iTj =
with equality if and only if D1 þ DT 11 ¼ D2 þ DT 22 ¼ ¼ Dn þ DT nn . Proof. Note that DQ ðGÞ ¼ TrðGÞ þ DðGÞ, apply Theorem 2.4 to DQ ðGÞ. Since aij ¼ dij ; r i ¼ Di and si ¼ T i for i ¼ 1; 2; . . . ; n, we have the desired upper bound for dQ1 ðGÞ. h
331
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
Theorem 3.8. Let G be a simple connected graph on n vertices, fD1 ; D2 ; . . . ; Dn g be the transmission sequence of G with D1 P D2 P P Dn and d be the diameter of G. Then for i ¼ 1; 2; . . . ; n,
dQ1 ðGÞ
6
2Di þ D1 d þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2Di þ d D1 Þ þ 8ði 1ÞðD1 Di Þd 2
:
Moreover, if i ¼ 1, the equality holds if and only if G is a transmission regular graph; and if 2 6 i 6 n, the equality holds if and only if G ffi K n . Proof. Apply Theorem 2.9 to DQ ðGÞ. Since M ¼ D1 ; N ¼ d and r i ¼ 2Di for i ¼ 1; 2; . . . ; n, we have the desired upper bound for dQ1 ðGÞ. If i ¼ 1, the equality holds if and only if G is a transmission regular graph; and if 2 6 i 6 n, the equality holds if and only if G ffi K n by the conditions (i)–(iii) in Theorem 2.9. h Theorem 3.9. Let G be a simple connected graph on n vertices, fD1 ; D2 ; . . . ; Dn g be the transmission sequence of G with D1 P D2 P P Dn and fT 1 ; T 2 ; . . . ; T n g be the second distance degree sequence of G. Then
Ti Ti min Di þ : 1 6 i 6 n 6 dQ1 ðGÞ 6 max Di þ : 1 6 i 6 n : Di Di
Moreover, any equality holds if and only if G has the same value Di þ DT ii for all i. Proof. Let DQ ðGÞ ¼ ðqij Þ, where qij ¼
Di ; dij ;
if i ¼ j; q D Then for any i; j 2 f1; 2; . . . ; ng, the ði; jÞ–entry of TrðGÞ1 DQ ðGÞTrðGÞ is ijD j i if i – j:
and the row sum of TrðGÞ1 DQ ðGÞTrðGÞ is ri ðTrðGÞ1 DQ ðGÞTrðGÞÞ ¼ Di þ DT i for i ¼ 1; 2; . . . ; n. i
It is easy to show that TrðGÞ1 DQ ðGÞTrðGÞ is a nonnegative and irreducible n n matrix with spectral radius dQ1 ðGÞ. By Lemma 2.1, the desired results hold. h Theorem 3.10. Let G be a simple connected graph on n (n P 2) vertices, fD1 ; D2 ; . . . ; Dn g be the transmission sequence of G with D1 P D2 P P Dn and fT 1 ; T 2 ; . . . ; T n g be the second distance degree sequence of G. Then
min
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2T i þ 2D2i : 1 6 i 6 n 6 dQ1 ðGÞ 6 max 2T i þ 2D2i : 1 6 i 6 n ;
and each equality holds if and only if G has the same value T i þ D2i for all i. 2
Proof. Let DQ ðGÞ ¼ ðqij Þ, then ððDQ ðGÞÞ Þij ¼ 2
ri ððDQ ðGÞÞ Þ ¼
Pn
k¼1 qik qkj
2
and the row sum of ðDQ ðGÞÞ is
n X n n n n X X X X qik qkj ¼ qik qkj ¼ 2 qik Dk ¼ 2T i þ 2D2i : j¼1 k¼1
k¼1
j¼1
k¼1 2
2
Let x be the unit Perron vector corresponding to dQ1 ðGÞ. Then DQ ðGÞx ¼ dQ1 ðGÞx and ðDQ ðGÞÞ x ¼ ðdQ1 ðGÞÞ x. By Lemma 2.1,
2
2 min 2T i þ 2D2i : 1 6 i 6 n 6 dQ1 ðGÞ ¼ d1 ððDQ ðGÞÞ Þ 6 max 2T i þ 2D2i : 1 6 i 6 n : Thus min
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2T i þ 2D2i : 1 6 i 6 n 6 dQ1 ðGÞ 6 max 2T i þ 2D2i : 1 6 i 6 n . 2
Note that G is connected graph on n (n P 2) vertices, then ðDQ ðGÞÞ is irreducible. We assume that G has the same value T i þ D2i for all i. Thus
min
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2T i þ 2D2i : 1 6 i 6 n ¼ max 2T i þ 2D2i : 1 6 i 6 n :
Both of the equalities hold.
2 2 Conversely, if one of the equalities holds, that is, ðdQ1 ðGÞÞ ¼ min 2T i þ 2D2i : 1 6 i 6 n or ðdQ1 ðGÞÞ ¼ max
2 2 Q 2 2T i þ 2Di : 1 6 i 6 n , by Lemma 2.1, ri ððD ðGÞÞ Þ ¼ 2T i þ 2Di , where 1 6 i 6 n, are all equal. Thus G has the same value T i þ D2i for all i. h Acknowledgments The authors would like to thank the referees for their valuable comments, corrections and suggestions, which lead to an improvement of the original manuscript.
332
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
References [1] [2] [3] [4] [5] [6]
M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013) 21–33. S. Burcu Bozkurt, Durmus Bozkurt, On the signless Laplacian spectral radius of digraphs, Ars Combin. 108 (2013) 193–200. K.C. Das, Maximizing the sum of the squares of the degree of a graph, Discrete Math. 285 (2004) 57–66. P. Hansen, C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl. 432 (2010) 3319–3336. C.X. He, Y. Liu, Z.H. Zhao, Some new sharp bounds on the distance spectral radius of graph, MATCH Commun. Math. Comput. Chem. 63 (2010) 783–788. A.D. Maden, K.C. Das, A.S. Çevik, Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph, Appl. Math. Comput. 219 (2013) 5025–5032. [7] H. Minc, Nonnegative Matrices, John Wiley and Sons Inc., New York, 1988. [8] I. Schur, Uber eine klasse von mittelbildungen mit anwendungen die determinanten, Sitzungsber. Berlin Math. Ges. 22 (1923) 9–20. [9] G.L. Yu, Y.R. Wu, J.L. Shu, Sharp bounds on the signless Laplacian spectral radii of graphs, Linear Algebra Appl. 434 (2011) 683–687.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Further results on the spectral radius of matrices and graphs q Wenxi Hong, Lihua You ⇑ School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China
a r t i c l e
i n f o
Keywords: Distance Signless Laplacian Spectral radius Bounds Graph
a b s t r a c t In this paper, we obtain some results on the sharp upper bounds of the spectral radius of a nonnegative matrix, then apply these results to signless Laplacian matrices to obtain some known results. We also apply these results to distance signless Laplacian matrix of a graph, and obtain some sharp bounds on the distance signless Laplacian spectral radius. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Let A ¼ ðaij Þ be an n n nonnegative matrix. The spectral radius of A, denoted by kðAÞ, is the largest modulus of eigenvalues of A. Furthermore, if A is irreducible, then kðAÞ is just the largest eigenvalue of A. Let G be a simple graph with vertex set VðGÞ ¼ fv 1 ; v 2 ; . . . ; v n g. Let AðGÞ ¼ ðaij Þnn be the ð0; 1Þ–adjacency matrix of G, where aij ¼ 1 if v i and v j are adjacent and 0 otherwise. Let di be the degree of the vertex v i in G for i ¼ 1; 2; . . . ; n and satisfy d1 P d2 P P dn and DðGÞ ¼ diagðd1 ; d2 ; . . . ; dn Þ be the degree diagonal matrix. Then Q ðGÞ ¼ DðGÞ þ AðGÞ is the signless Laplacian matrix of G. Clearly, Q ðGÞ is symmetric. Let q1 ðGÞ be the spectral radius of Q ðGÞ and it is also called the signless Laplacian spectral radius of G. The distance matrix DðGÞ ¼ ðdij Þ of G, its ði; jÞ–entry, dij , is equal to dG ðv i ; v j Þ which denotes the distance (the length of the shortest path) between vertex v i and vertex v j . Clearly, the distance matrix of a connected graph is irreducible and symmetric. The eigenvalues of DðGÞ are given as d1 ðGÞ P d2 ðGÞ P P dn ðGÞ, where d1 is called the distance spectral radius of G. The transmission of v i is defined to be the sum of the distance from v i to all other vertices in G, denoted by Di , that is, P Di ¼ nj¼1 dij . The transmission is also called the first distance degree [5]. Assume that the transmissions are ordered as D1 P D2 P P Dn . A connected graph G is said to be k–transmission regular if Di ¼ k for all i 2 f1; 2; . . . ; ng. Let TrðGÞ ¼ diagðD1 ; D2 ; . . . ; Dn Þ denote the diagonal matrix of its vertex transmissions. Aouchiche and Hansen [1] introduced the Laplacian and the signless Laplacian for the distance matrix of a connected graph. The matrix DL ðGÞ ¼ TrðGÞ DðGÞ is called the distance Laplacian of G, while the matrix DQ ðGÞ ¼ TrðGÞ þ DðGÞ is called the distance signless Laplacian of G. Let G be a connected graph, then the matrix DQ ðGÞ ¼ ðqij Þ is symmetric, nonnegative and irreducible. All the eigenvalues of DQ ðGÞ can be arranged as: dQ1 ðGÞ P dQ2 ðGÞ P P dQn ðGÞ. dQ1 ðGÞ is called the distance signless Laplacian spectral radius of G. As the DQ ðGÞ is irreducible, by the Perron–Frobenius theorem, dQ1 ðGÞ is positive, simple and there is a unique positive unit eigenvector x corresponding to dQ1 ðGÞ, which is called the distance signless Perron vector of G.
q Research supported by the Zhujiang Technology New Star Foundation of Guangzhou (No. 2011J2200090), and Program on International Cooperation and Innovation, Department of Education, Guangdong Province (No. 2012gjhz0007). ⇑ Corresponding author. E-mail address: [email protected] (L. You).
http://dx.doi.org/10.1016/j.amc.2014.04.107 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
327
As usual, we denote by K n the complete graph, by C n the cycle and by K a;na the complete bipartite graph. In [1], Aouchiche and Hansen obtain the lower sharp bound as dQ1 ðGÞ P dQ1 ðK n Þ ¼ 2n 2. In Section 2, we obtain some results on the sharp upper bounds of the spectral radius of a nonnegative matrix, and apply these results to the signless Laplacian matrix to obtain some known results. In Section 3, we present some sharp bounds on the distance signless Laplacian spectral radius of graphs. 2. Further results on the spectral radius of nonnegative matrices In this section, we obtain some new results on the sharp upper bounds of the spectral radius of nonnegative matrices. Then we apply these results to the signless Laplacian matrix and obtain some known results. Lemma 2.1 [7]. If A is an n n nonnegative matrix with the spectral radius kðAÞ and row sums r 1 ; r 2 ; . . . ; rn , then min16i6n r i 6 kðAÞ 6 max16i6n r i . Moreover, if A is irreducible, then one of the equalities holds if and only if the row sums of A are all equal. By Lemma 2.1 and the Cauchy–Weil inequality, we can get the following proposition. Proposition 2.2. Let A be an n n nonnegative, irreducible and symmetric matrix with row sums r 1 ; r2 ; . . . ; r n , where r 1 P r 2 P . . . P r n ; B ¼ A þ M where M ¼ diagðr1 ; r2 ; . . . ; r n Þ, and kðAÞ ðkðBÞÞ the spectral radius of A ðBÞ. Then kðBÞ 6 kðAÞ þ r 1 with equality if and only if the row sums of A are all equal. Let G be a simple and connected graph. Let A be the adjacency matrix of G and M be the degree diagonal matrix of G. Then B ¼ A þ M is the signless Laplacian matrix of G. So we can get the following corollary by Proposition 2.2. Corollary 2.3 [4, Lemma 9]. Let G be a simple and connected graph on n vertices, k1 ðGÞ be the spectral radius of G; q1 ðGÞ be the signless Laplacian spectral radius of G and D be the maximum degree of G. Then q1 ðGÞ 6 k1 ðGÞ þ D with equality if and only if G is regular. Theorem 2.4. Let A ¼ ðaij Þ be an n n nonnegative and irreducible matrix with aii ¼ 0 for i ¼ 1; 2; . . . ; n and row sums P r 1 ; r2 ; . . . ; r n with r 1 P r2 P P r n . Let B ¼ A þ M, where M ¼ diagðr1 ; r2 ; . . . ; r n Þ, si ¼ nj¼1 aij r j for any i 2 f1; 2; . . . ; ng and kðBÞ be the spectral radius of B. Then
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8
ð2:1Þ
with equality if and only if r 1 þ rs11 ¼ r 2 þ rs22 ¼ ¼ rn þ rsnn .
r i ; if i ¼ j; then B and M 1 BM are nonnegative and aij ; if i – j; 1 irreducible, and M BM and B have the same eigenvalue. Let x ¼ ðx1 ; x2 ; . . . ; xn ÞT be a positive eigenvector of M 1 BM corresponding to kðBÞ. We assume that one entry, say xp , is equal to 1 and the others are less than or equal to 1, that is, xp ¼ 1 and 0 < xk 1 for any k. Let xq ¼ maxfxk j1 6 k 6 n and k – pg. From M 1 BMx ¼ kðBÞx, we have Proof. Since A is irreducible, r i > 0, and B ¼ ðbij Þ, where bij ¼
kðBÞ ¼ kðBÞxp ¼
n X bpk r k xk k¼1
rp
¼ rp þ
n n X X bpk r k xk apk r k xk ¼ rp þ ; rp rp k¼1;k–p k¼1;k–p
thus
kðBÞ ¼ r p þ
n n X 1X 1 2 apk r k xk 6 r p þ xq apk r k rp k¼1 rp k¼1
! ¼ rp þ
xq s p ; rp
ð2:2Þ
with equality if and only if xk ¼ xq for all k 2 f1; 2; . . . ; ng n fpg, and
kðBÞxq ¼
n X bqk r k xk k¼1
rq
¼ r q xq þ
n n 1X 1X sq aqk r k xk 6 rq xq þ aqk r k ¼ r q xq þ ; r q k¼1 r q k¼1 rq
ð2:3Þ
with equality if and only if xk ¼ xp ¼ 1 for all k 2 f1; 2; . . . ; ng n fqg. From (2.2) and (2.3), we get n 1X xq s p apk r k xk ¼ kðBÞ rp 6 ; rp k¼1 rp
ð2:4Þ
n 1X sq aqk r k xk ¼ ðkðBÞ r q Þxq 6 : rq k¼1 rq
ð2:5Þ
and
328
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
From (2.4) and (2.5), we get kðBÞ r p > 0 and kðBÞ r q > 0 easily, and
k2 ðBÞ ðr p þ r q ÞkðBÞ þ r p r q
sp sq 6 0: rp rq
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Thus we have kðBÞ 6
r p þr q þ
4sp sq q
ðr p r q Þ2 þ rp r
i j
, then (2.1) holds. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8
6 max16i;j6n
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8
:
;
() x1 ¼ x2 ¼ ¼ xn ¼ 1 () ðM 1 BMÞð1; 1; . . . ; 1ÞT ¼ kðBÞð1; 1; . . . ; 1ÞT () BðMð1; 1; . . . ; 1ÞT Þ ¼ kðBÞðMð1; 1; . . . ; 1ÞT Þ () Bðr1 ; r2 ; . . . ; rn ÞT ¼ kðBÞðr 1 ; r2 ; . . . ; r n ÞT P P () r 2i þ si ¼ r 2i þ nj¼1 aij rj ¼ r2i þ nj¼1;j–i aij r j ¼ kðBÞri for any i 2 f1; 2; . . . ; ng () kðBÞ ¼ r i þ rsi for any i 2 f1; 2; . . . ; ng: i
) r 1 þ sr11 ¼ r2 þ rs22 ¼ ¼ r n þ rsnn : Now we show if r 1 þ rs11 ¼ r 2 þ rs22 ¼ ¼ r n þ rsnn , then kðBÞ ¼ r i þ rsii for any i 2 f1; 2; . . . ; ng. Let k ¼ r1 þ rs11 ¼ r 2 þ rs22 ¼ ¼ r n þ rsnn , then T
Bðr 1 ; r 2 ; . . . ; r n ÞT ¼ ðr 21 þ s1 ; r 22 þ s2 ; . . . ; r 2n þ sn Þ ¼ kðr 1 ; r 2 ; . . . ; r n ÞT ; it implies that k is an eigenvalue of B, so k 6 kðBÞ. 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9
P
v k v i dk be the average degree of the neighbors of
v i , where vertices v i and v k
are adjacent in G.
Lemma 2.5 [3]. Let G be a connected graph. For v i 2 VðGÞ, the degree of v i and the average degree of the vertices adjacent to v i are denoted by di and mi , respectively. Then d1 þ m1 ¼ d2 þ m2 ¼ ¼ dn þ mn holds if and only if G is a regular graph or a bipartite semiregular graph. We apply Theorem 2.4 to the signless Laplacian matrix Q ðGÞ ¼ DðGÞ þ AðGÞ of Q, then we can obtain the following corollary easily by Lemma 2.5. Corollary 2.6 [6, Theorem 6(b)]. Let G be a simple and connected graph on n vertices with degrees sequence fd1 ; d2 ; . . . ; dn g P which satisfies d1 P d2 P P dn and q1 ðGÞ be the signless Laplacian spectral radius of G. Let si ¼ v k v i dk , where vertices v i and v k are adjacent, for i 2 f1; 2; . . . ; ng. Then
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 i sj
bors of the vertex
i
v i . Let AðDÞ ¼ ðaij Þ denote the adjacency matrix of D, where aij
is equal to the number of arc ðv i ; v j Þ. We þ
þ
þ
apply Theorem 2.4 to the signless Laplacian matrix Q ðDÞ of a digraph D, where Q ðDÞ ¼ diagðd1 ; d2 ; . . . ; dn Þ þ AðDÞ. Then we can obtain the following corollary easily. Corollary 2.7 [2, Theorem 3.2]. Let D be a strongly connected digraph on n vertices. Then
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 2 > > þ þ þ þ > > þ 4mþi mþj di dj
conclude
that
the
equality
in
Corollary
2.7
holds
if
and
only
if
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W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
Theorem 2.9. Let A ¼ ðaij Þ be an n n nonnegative matrix with row sums r1 ; r2 ; . . . ; r n , where r 1 P r2 P P r n ; M be the largest diagonal element, N be the largest non-diagonal element and kðAÞ be the spectral radius of A. Then for i ¼ 1; 2; . . . ; n,
kðAÞ 6
ri þ M N þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðri þ N MÞ2 þ 4ði 1Þðr1 ri ÞN 2
ð2:6Þ
:
Moreover, if A is irreducible, when i ¼ 1, then the equality holds if and only if the row sums of A are all equal; when 2 6 i 6 n, then the equality holds if and only if A satisfies the following conditions: (i) all ¼ M for 1 6 l 6 i 1; (ii) alk ¼ N for 1 6 l; k 6 n and l – k; (iii) r1 ¼ r2 ¼ ¼ r i1 P r i ¼ r iþ1 ¼ ¼ r n . Proof. When i ¼ 1 or r 1 ¼ r i , it is clearly that the inequality is true and the equality holds if and only if the row sums of A are all equal by Lemma 2.1. We suppose r1 > r i and 2 6 i 6 n. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 x; 1 6 j 6 i 1; i þNMÞ þ4ði1Þðr 1 r i ÞN Let x ¼ Mri þð2i3ÞNþ ðr2ði1ÞN . It is easy to show that x > 1. Let xj ¼ Let 1; i 6 j 6 n: 1 U ¼ diagðx1 ; x2 ; . . . ; xn Þ be a diagonal matrix of order n. Let B ¼ U AU and note that B and A have the same eigenvalues, then kðBÞ ¼ kðAÞ. Now we consider the row sums s1 ; s2 ; . . . ; sn of matrix B. For 1 6 l 6 i 1, we have
sl ¼
n X xj j¼1
xl
alj ¼
i1 i1 i1 n n X 1X 1X 1 X 1 1 X alj þ alj ¼ alj þ 1 alj ¼ r l þ 1 alj ; x j¼i x j¼1 x j¼1 x x j¼1 j¼1
and for i 6 l 6 n,
sl ¼
n X xj j¼1
xl
alj ¼
n i1 i1 X X X alj þ ðx 1Þ alj ¼ rl þ ðx 1Þ alj : j¼1
j¼1
j¼1
Since A is a nonnegative matrix, then 0 6 apq 6 N if p – q, where 1 6 p; q 6 n. Thus 1 6 l 6 i 1, and
Pi1
j¼1 alj
6 M þ ði 2ÞN, where
i1 X alj 6 ði 1ÞN; where i 6 l 6 n: j¼1
As x > 1 and r1 P r 2 P P rn , then for 1 6 l 6 i 1,
sl 6
1 1 1 1 1 1 ½M þ ði 2ÞN ¼ ðrl MÞ þ M þ 1 ði 2ÞN 6 ðr 1 MÞ þ M þ 1 ði 2ÞN; rl þ 1 x x x x x x
with equality if and only if ðaÞ and ðbÞ hold: ðaÞall ¼ M and alj ¼ N if 1 6 j 6 i 1 with j – i; ðbÞr 1 ¼ rl . For i 6 l 6 n, we have sl 6 rl þ ðx 1Þði 1ÞN 6 r i þ ðx 1Þði 1ÞN, with equality if and only if ðcÞ and ðdÞ hold: ðcÞalj ¼ N if 1 6 j 6 i 1; ðdÞri ¼ rl . Noting that
M þ ri N þ 1 1 ði 2ÞN ¼ ri þ ðx 1Þði 1ÞN ¼ ðr1 MÞ þ M þ 1 x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
Mþr Nþ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr i þ N MÞ2 þ 4ði 1Þðr 1 r i ÞN 2
;
ðr þNMÞ þ4ði1Þðr r ÞN
1 i i i thus kðAÞ ¼ kðBÞ 6 maxfs1 ; s2 ; . . . ; sn g 6 . 2 When equality holds in (2.6), then ðaÞ ðbÞ hold for 1 6 l 6 i 1; ðcÞ ðdÞ hold for i 6 l 6 n. Thus all ¼ M for 1 6 l 6 i 1, r 1 ¼ r 2 ¼ ¼ r i1 > ri ¼ r iþ1 ¼ ¼ r n and all the non-diagonal elements are equal. Now (i)–(iii) follow. Conversely, if (i)–(iii) hold, it is easy to show that equality holds. h
Similarly, we apply Theorem 2.9 to Q ðGÞ ¼ DðGÞ þ AðGÞ and note that r i ¼ 2di for any i 2 f1; 2; . . . ; ng; M ¼ d1 and N ¼ 1. Then we have Corollary 2.10 [9, Theorem 3.2]. Let G be a simple and connected graph with n vertices and degrees sequence fd1 ; d2 ; . . . ; dn g which satisfies d1 P d2 P P dn and q1 ðGÞ be the signless Laplacian spectral radius of G. Then for i ¼ 1; 2; . . . ; n,
q1 ðGÞ 6
2di þ d1 1 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2di þ 1 d1 Þ þ 8ði 1Þðd1 di Þ 2
:
330
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
Moreover, if i ¼ 1, then the equality holds if and only if G is a regular graph; if 2 6 i 6 n, then the equality holds if and only if G is either a regular graph or a bidegreed graph in which d1 ¼ d2 ¼ ¼ di1 ¼ n 1 and di ¼ diþ1 ¼ ¼ dn . 3. Bounds on distance signless Laplacian spectral radius In this section, we present some sharp bounds on the distance signless Laplacian spectral radius by the above theorems and lemmas in Section 2. P Noting that the ith row sum of DðGÞ (DQ ðGÞ) is ri ¼ Di (ri ¼ Di þ nj¼1 dij ¼ 2Di ), we can obtain the following results by Lemma 2.1 easily. Proposition 3.1. Let G be a simple connected graph on n vertices, D1 and Dn be the largest and least transmissions of G, respectively. Then Dn 6 d1 ðGÞ 6 D1 . Moreover, one of the equalities holds if and only if G is a transmission regular graph.
Proposition 3.2. Let G be a simple connected graph on n vertices, D1 and Dn be the largest and least transmissions of G, respectively. Then 2Dn 6 dQ1 ðGÞ 6 2D1 . Moreover, one of the equalities holds if and only if G is a transmission regular graph. By Proposition 3.2, we obtain the largest distance signless Laplacian spectral radius of K n ; K n2;n2 and C n as follows.
dQ1 ðK n Þ
¼ 2ðn 1Þ;
dQ1
K 2n;2n ¼ 3n 4 and dQ1 ðC n Þ ¼
(
n2 1 ; 2 n2 2
;
if n is odd; if n is even:
Lemma 3.3 [1]. Let G be a connected graph on nðP 2Þ vertices. Then dQ1 ðGÞ P dQ1 ðK n Þ ¼ 2ðn 1Þ and dQi ðGÞ P dQi ðK n Þ ¼ n 2, for all 2 6 i 6 n. Moreover, dQ2 ðGÞ ¼ dQ2 ðK n Þ ¼ n 2 if and only if G is the complete graph K n .
Lemma 3.4 [8]. Let A be a positive semidefinite Hermitian matrix, k1 ; k2 ; . . . ; kn be the eigenvalues of A where k1 P k2 P P kn and a1 ; a2 ; . . . ; an be the entries of the main diagonal of A satisfying a1 P a2 P P an . Then the spectrum of A majorizes its main P P P P diagonal, that is, ki¼1 ki P ki¼1 ai for k ¼ 1; 2; . . . ; n, and ni¼1 ki ¼ ni¼1 ai . Theorem 3.5. Let G be a simple connected graph on n vertices, d1 ðGÞ be the distance spectral radius of G and D1 be the largest transmission of G. Then D1 6 dQ1 ðGÞ 6 d1 ðGÞ þ D1 . Moreover, dQ1 ðGÞ ¼ d1 ðGÞ þ D1 if and only if G is a transmission regular graph. Proof. Denote the spectrum of DQ ðGÞ by spðDQ ðGÞÞ ¼ ðdQ1 ðGÞ; dQ2 ðGÞ; . . . ; dQn ðGÞÞ and the transmissions D1 ; D2 ; . . . ; Dn of G satisfy D1 P D2 P P Dn . Noting that dQn ðGÞ P dQn ðK n Þ ¼ n 2 P 0 for n P 2 by Lemma 3.3, then DQ ðGÞ is a positive semidefinite Hermitian matrix. By Lemma 3.4, we have dQ1 ðGÞ P D1 . By Proposition 2.2, we directly get the dQ1 ðGÞ 6 d1 ðGÞ þ D1 with equality if and only if G is a transmission regular graph. h
Definition 3.6 [5]. Let G be a simple connected graph on n vertices, DðGÞ ¼ ðdij Þ be the distance matrix and the transmission sequence fD1 ; D2 ; . . . ; Dn g with D1 P D2 P P Dn . Then the second distance degree of v i , denoted by T i , is given by P T i ¼ ni¼1 dij Di . We can easily obtain Theorems 3.7 and 3.8 from Theorems 2.4 and 2.9, respectively. Theorem 3.7. Let G be a simple connected graph on n vertices, fD1 ; D2 ; ; Dn g be the transmission sequence of G with D1 P D2 P P Dn and fT 1 ; T 2 ; . . . ; T n g be the second distance degree sequence of G. Then
dQ1 ðGÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 iTj =
with equality if and only if D1 þ DT 11 ¼ D2 þ DT 22 ¼ ¼ Dn þ DT nn . Proof. Note that DQ ðGÞ ¼ TrðGÞ þ DðGÞ, apply Theorem 2.4 to DQ ðGÞ. Since aij ¼ dij ; r i ¼ Di and si ¼ T i for i ¼ 1; 2; . . . ; n, we have the desired upper bound for dQ1 ðGÞ. h
331
W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
Theorem 3.8. Let G be a simple connected graph on n vertices, fD1 ; D2 ; . . . ; Dn g be the transmission sequence of G with D1 P D2 P P Dn and d be the diameter of G. Then for i ¼ 1; 2; . . . ; n,
dQ1 ðGÞ
6
2Di þ D1 d þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2Di þ d D1 Þ þ 8ði 1ÞðD1 Di Þd 2
:
Moreover, if i ¼ 1, the equality holds if and only if G is a transmission regular graph; and if 2 6 i 6 n, the equality holds if and only if G ffi K n . Proof. Apply Theorem 2.9 to DQ ðGÞ. Since M ¼ D1 ; N ¼ d and r i ¼ 2Di for i ¼ 1; 2; . . . ; n, we have the desired upper bound for dQ1 ðGÞ. If i ¼ 1, the equality holds if and only if G is a transmission regular graph; and if 2 6 i 6 n, the equality holds if and only if G ffi K n by the conditions (i)–(iii) in Theorem 2.9. h Theorem 3.9. Let G be a simple connected graph on n vertices, fD1 ; D2 ; . . . ; Dn g be the transmission sequence of G with D1 P D2 P P Dn and fT 1 ; T 2 ; . . . ; T n g be the second distance degree sequence of G. Then
Ti Ti min Di þ : 1 6 i 6 n 6 dQ1 ðGÞ 6 max Di þ : 1 6 i 6 n : Di Di
Moreover, any equality holds if and only if G has the same value Di þ DT ii for all i. Proof. Let DQ ðGÞ ¼ ðqij Þ, where qij ¼
Di ; dij ;
if i ¼ j; q D Then for any i; j 2 f1; 2; . . . ; ng, the ði; jÞ–entry of TrðGÞ1 DQ ðGÞTrðGÞ is ijD j i if i – j:
and the row sum of TrðGÞ1 DQ ðGÞTrðGÞ is ri ðTrðGÞ1 DQ ðGÞTrðGÞÞ ¼ Di þ DT i for i ¼ 1; 2; . . . ; n. i
It is easy to show that TrðGÞ1 DQ ðGÞTrðGÞ is a nonnegative and irreducible n n matrix with spectral radius dQ1 ðGÞ. By Lemma 2.1, the desired results hold. h Theorem 3.10. Let G be a simple connected graph on n (n P 2) vertices, fD1 ; D2 ; . . . ; Dn g be the transmission sequence of G with D1 P D2 P P Dn and fT 1 ; T 2 ; . . . ; T n g be the second distance degree sequence of G. Then
min
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2T i þ 2D2i : 1 6 i 6 n 6 dQ1 ðGÞ 6 max 2T i þ 2D2i : 1 6 i 6 n ;
and each equality holds if and only if G has the same value T i þ D2i for all i. 2
Proof. Let DQ ðGÞ ¼ ðqij Þ, then ððDQ ðGÞÞ Þij ¼ 2
ri ððDQ ðGÞÞ Þ ¼
Pn
k¼1 qik qkj
2
and the row sum of ðDQ ðGÞÞ is
n X n n n n X X X X qik qkj ¼ qik qkj ¼ 2 qik Dk ¼ 2T i þ 2D2i : j¼1 k¼1
k¼1
j¼1
k¼1 2
2
Let x be the unit Perron vector corresponding to dQ1 ðGÞ. Then DQ ðGÞx ¼ dQ1 ðGÞx and ðDQ ðGÞÞ x ¼ ðdQ1 ðGÞÞ x. By Lemma 2.1,
2
2 min 2T i þ 2D2i : 1 6 i 6 n 6 dQ1 ðGÞ ¼ d1 ððDQ ðGÞÞ Þ 6 max 2T i þ 2D2i : 1 6 i 6 n : Thus min
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2T i þ 2D2i : 1 6 i 6 n 6 dQ1 ðGÞ 6 max 2T i þ 2D2i : 1 6 i 6 n . 2
Note that G is connected graph on n (n P 2) vertices, then ðDQ ðGÞÞ is irreducible. We assume that G has the same value T i þ D2i for all i. Thus
min
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2T i þ 2D2i : 1 6 i 6 n ¼ max 2T i þ 2D2i : 1 6 i 6 n :
Both of the equalities hold.
2 2 Conversely, if one of the equalities holds, that is, ðdQ1 ðGÞÞ ¼ min 2T i þ 2D2i : 1 6 i 6 n or ðdQ1 ðGÞÞ ¼ max
2 2 Q 2 2T i þ 2Di : 1 6 i 6 n , by Lemma 2.1, ri ððD ðGÞÞ Þ ¼ 2T i þ 2Di , where 1 6 i 6 n, are all equal. Thus G has the same value T i þ D2i for all i. h Acknowledgments The authors would like to thank the referees for their valuable comments, corrections and suggestions, which lead to an improvement of the original manuscript.
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W. Hong, L. You / Applied Mathematics and Computation 239 (2014) 326–332
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