On the distance signless Laplacian spectral radius of graphs and digraphs

Electronic Journal of Linear Algebra Volume 3 Volume 3 (017) Article 3 017 On the distance signless Laplacian spectral radius of graphs and digraphs Dan Li Xinjiang University,Urumqi, ldxjedu@163.com Guoping Wang Xinjiang Normal University,Urumqi, xj.wgp@163.com Jixiang Meng Xinjiang University, Urumqi, mjxxju@sina.com Follow this and additional works at: Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Li, Dan; Wang, Guoping; and Meng, Jixiang. (017), "On the distance signless Laplacian spectral radius of graphs and digraphs", Electronic Journal of Linear Algebra, Volume 3, pp. 438-446. DOI: https://doi.org/10.13001/1081-3810.198 This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact scholcom@uwyo.edu.

Volume 3, pp. 438-446, December 017. ON THE DISTANCE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS AND DIGRAPHS DAN LI, GUOPING WANG, AND JIXIANG MENG Abstract. Let η(g) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper, bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized. Key words. Distance signless Laplacian spectral radius, Bound, Extremal graph, Digraph, Vertex connectivity. AMS subject classifications. 05C50. 1. Introduction. In this paper, we consider simple and connected graphs. Let G be a simple graph with vertex set V (G) = {v 1, v,..., v n } and edge set E(G). Let d i be the degree of the vertex v i in G for i = 1,..., n. Let A(G) = (a ij ) n n be the (0, 1)-adjacency matrix of G, where a ij = 1 if v i v j E(G) and a ij = 0 otherwise. Let (G) = diag(d 1,..., d n ) be the diagonal degree matrix. Then Q(G) = (G) + A(G) is the signless Laplacian matrix of G. Let D(G) = (d ij ) n n be the distance matrix of a connected graph G, where d ij = d G (v i, v j ) is defined to be the length of shortest path between v i and v j. We call D i = n j=1 d ij the transmission of vertex v i (i = 1,,..., n). The matrix D Q (G) = Diag(T r) + D(G) is the distance signless Laplacian matrix of G (D Q -matrix), where Diag(T r) is the diagonal matrix whose ith diagonal entry is D i. The matrix D Q (G) is nonnegative and irreducible when G is connected. The largest eigenvalue of D Q (G) is called distance signless Laplacian spectral radius of the graph G. The distance spectral radius of a connected graph has been studied extensively. Chen, Lin and Shu [4] obtained sharp upper bounds on the distance spectral radius of a graph. Indulal [7] gave some bounds for the distance spectral radius of graphs. Ilić [6] obtained the tree with given matching number that minimizes the distance spectral radius. Bose, Nath and Paul [3] determined the unique graph with maximal distance spectral radius in the class of graphs without a pendent vertex. Lin, Yang, Zhang and Shu [10] characterized the extremal digraph with the minimal distance spectral radius among all digraphs with given vertex connectivity and the extremal graph with the minimal distance spectral radius among all graphs with given edge connectivity. Aouchiche and Hansen [1] introduced the distance Laplacian and distance signless Laplacian spectra of graphs, respectively. Lin and Lu [9] found a sharp lower bound as well as a sharp upper bound of the distance signless Laplacian spectral radius in terms of the clique number. Furthermore, both extremal graphs are Received by the editors on June 7, 017. Accepted for publication on November 13, 017. Handling Editor: Ravi Bapat. College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China (ldxjedu@163.com, mjxxju@sina.com). School of Mathematical Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830054, China (xj.wgp@163.com).

Volume 3, pp. 438-446, December 017. 439 On the Distance Signless Laplacian Spectral Radius of Graphs and Digraphs uniquely determined. Hong and You [5] obtained some sharp bounds on the distance signless Laplaciqan spectral radius of graphs. Xing and Zhou [11] gave the unique graphs with minimum distance and distance signless Laplacian spectral radius among bicyclic graphs with fixed number of vertices. Xing, Zhou and Li [1] determined the graphs with minimum distance signless Laplacian spectral radius among the trees, unicyclic graphs, bipartite graphs, the connected graphs with fixed pendant vertices and fixed connectivity, respectively. In this paper, we obtain the bounds for the distance signless Laplacian spectral radius of graphs and determine the extremal graph with the minimum distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree. Furthermore, we characterize the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity.. Bounds on the distance signless Laplacian spectral radius of graphs. Let G be a graph. We denote the distance signless Laplacian spectral radius of G by η(g). A graph G is transmission regular if D 1 = D = = D n. Lemma.1. [5] Let G be a connected graph on n vertices. Then min{d i : 1 i n} η(g) max{d i : 1 i n}. Moreover, one of the equalities holds if and only if G is a transmission regular graph. Let G be a graph on n vertices. It was shown in [11] that η(g) n G is transmission regular. Note that 1 n ( n Di D i ) D1. Then +D + +D n n n better lower bound of η(g). Theorem.. Let G be a graph on n vertices. Then η(g) if G is transmission regular. Proof. Let x = 1 n (1, 1,..., 1). Then η(g) x(d Q (G)) x T. D i with equality if and only if D i. Thus, the following result gives a D 1 +D + +D n n with equality if and only Note that xd Q (G) = 1 n (1, 1,..., 1)D Q (G) = n (D 1, D,..., D n ). We have D1 Thus, η(g) +D + +D n n. x(d Q (G)) x T = xd Q (G)(xD Q (G)) T = 4 n If G is transmission regular, then by Lemma.1, η(g) = D i = Di. 1 n Di. Conversely, if equality holds, then x is the eigenvector corresponding to η(g), i.e., D Q (G)x T = η(g)x T. This implies that D i = η(g)/ (i = 1,,..., n).

Volume 3, pp. 438-446, December 017. Dan Li, Guoping Wang, and Jixiang Meng 440 Let ρ(g) be the distance spectral radius of the graph G. If G is transmission regular, then ρ(g) = D i. Thus, we have the following. Corollary.3. Suppose that G is transmission regular. Then η(g) = ρ(g). A graph G is regular if d 1 = = d n. Lemma.4. Let G be a connected graph with diameter d. Then G is regular if and only if G is transmission regular. Proof. If d = 1, then G = K n. If d =, then D i = d i + (n 1 d i ) = n d i, from which we can see that G is regular if and only if G is transmission regular. The unit eigenvector x(g) = (x 1, x,..., x n ) T > 0 corresponding to η(g) is called the Perron vector of D Q (G). Theorem.5. Let G be a graph on n vertices with maximum degree 1 and second maximum degree. Then η(g) 4n 4 1 with equality if and only if G is a regular graph with diameter d. Proof. Let x = (x 1, x,..., x n ) T be a Perron eigenvector of D Q (G) such that x i = min 1 k n x k and x j = min 1 k i n x k. From D Q (G)x = η(g)x, we obtain that Note that Then we can get that η(g)x i = D i x i + η(g)x j = D j x j + D i = D j = k=1,k i k=1,k j k=1,k i k=1,k j d ik x k D i (x i + x j ); d jk x k D j (x i + x j ). d ik d i + (n 1 d i ) = n d i ; d jk d j + (n 1 d j ) = n d j. (.1) η(g)x i (n d i )(x i + x j ) and (.) η(g)x j (n d j )(x i + x j ). Therefore, by (.1) and (.), we have that is η(g)(x i + x j ) (4n 4 d i d j )(x i + x j ) (4n 4 1 )(x i + x j ), η(g) 4n 4 1. If the equality holds, then we deduce that G is a regular graph with diameter d and all x i are equal. If d = 1, then G = K n. If d =, then we get η(g)x i = (d i + (n 1 d i ))x i, i.e., η(g) = (n d i ), which implies that G is a regular graph. Conversely, by Lemma.4, we can get that η(g) = (n 1 ) if G is a regular graph with diameter d.

Volume 3, pp. 438-446, December 017. 441 On the Distance Signless Laplacian Spectral Radius of Graphs and Digraphs Theorem.6. Let G be a graph on n vertices with minimum degree δ 1 and second minimum degree δ. Then η(g) dn d(d 1) (d 1)(δ 1 + δ ) with equality if and only if G is a regular graph with diameter d. Proof. Let x = (x 1, x,..., x n ) T be a Perron eigenvector of D Q (G) such that x s = max 1 k n x k and x t = max 1 k s n x k. From D Q (G)x = η(g)x, we obtain that Note that Thus, D s = k=1,k s η(g)x s = D s x s + η(g)x t = D t x t + k=1,k s k=1,k t d sk x k D s (x s + x t ); d tk x k D t (x s + x t ). d sk d s + + + d 1 + d[n 1 d s (d )] = dn (.3) η(g)x s [dn Similarly, we can get that (.4) η(g)x t [dn Therefore, by (.3) and (.4), we have d(d 1) d(d 1) 1 d s (d 1)](x s + x t ). 1 d t (d 1)](x s + x t ). d(d 1) η(g)(x s + x t ) [dn d(d 1) (d 1)(d s + d t )](x s + x t ), from which we obtain that η(g) dn d(d 1) (d 1)(δ 1 + δ ). 1 d s (d 1). If the equality holds, then all x i are equal, and hence, D Q (G) has equal row sums. Therefore, G is transmission regular. If d 3, then by (.3), we know that for each vertex v i, there is exactly one vertex v j such that d ij =, and then d < 4. If the diameter of G is 3 and equality holds, then for a center vertex v s, from D Q (G)x = η(g)x and (.3), written for the component x s, we have η(g)x s = D s x s + d s x s + (n 1 d s )x s = (n d s )x s = (3n 4 d s )x s, and thus, d s = n, which implies that G = P 4, but P 4 is not transmission regular, a contradiction. Therefore, G is a regular graph with diameter d. Conversely, by Lemma.4, we can get that η(g) = (n δ 1 ) if G is a regular graph with diameter d. 3. The minimal distance signless Laplacian spectral radius for graphs with given vertex connectivity and minimum degree. The vertex connectivity of a graph G is the minimum number of vertices, whose deletion results in a disconnected or a trivial graph. The join G 1 + G of two graphs G 1 (V 1, E 1 ) and G (V, E ) consists of the union G 1 G of G 1 and G and all edges joining V 1 with V. Let

Volume 3, pp. 438-446, December 017. Dan Li, Guoping Wang, and Jixiang Meng 44 G k,n be the class of all graphs of order n with vertex connectivity κ(g) k n 1 and minimum degree δ k. Let G k,δ,n = K k + (K δ k+1 K n δ 1 ). Obviously, G k,δ,n G k,n. For fixed n and k, let g(n 1, n ) = x 3 a 1 x a x + a 3, where a 1 = 4n + n 1 + n 6, a = (n 1 + n )(0 13k) 8(n + n 1) 1(n 1 n + 1) 5k + 16k, and a 3 = k 3 + (10 8n 8n 1 )k + ( 16 + 6n + 6n 1 10n 10n 1 16n 1 n )k 4((n ) + (n 1 ) + (n 1 + n )(n 1 n + 1) + (3n 1 n + 1)). Lemma 3.1. η = η(k k + (K n1 K n )) satisfies the equation g(n 1, n ) = 0. Proof. Let x be the Perron vector of D Q (K k + (K n1 K n ). It is clear that x can be written as From D Q (K k + (K n1 K n )x = ηx, we have x = (x,..., x, y,..., y, z,..., z) T. }{{}}{{}}{{} n 1 k n ηx = (n 1 + n + k )x + ky + n z; ηy = n 1 x + (n 1 + n + k )y + n z; ηz = n 1 x + ky + (n 1 + n + k )z. This shows that η is an eigenvalue of the matrix A, where n 1 + n + k k n A = n 1 n 1 + n + k n, n 1 k n 1 + n + k and det(ix A) = g(n 1, n ) = x 3 a 1 x a x + a 3. This shows that the result is true. By Lemma 3.1 we can immediately get the following corollary. Corollary 3.. Spectral radius of G k,δ,n is the largest root of the equation x 3 (5n k 6)x ( 4δ + (4k + 4n 8)δ nk 8n + 4n 16)x + (( 1k + 8n + 16)δ + (1k + 4kn 8n 40k + 3)δ 4k 3 kn 8k + 54kn 1n 56k + 4n 4) = 0. Lemma 3.3. If n n 1, then η(k k + (K n1 K n )) > η(k k + (K n1 1 K n+1)). Proof. Let h(λ) = g n1,n (λ) g n1 1,n +1(λ) = 4(n n 1 + 1)λ + 4(k(n n 1 + 1) + (n n 1 )(n + n 1 1) + (n 1 1)). Then λ 0 = k + (n n1)(n+n1 1) n n 1+1 + (n1 1) n n 1+1 is the root of the equation h(λ) = 0. By Lemma.1, we know that η(k k +(K n1 K n )) (n +n 1 1)+k and η(k k +(K n1 1 K n+1)) (n + n 1 1) + k. And, (n + n 1 1 + k) λ 0 = (n + n 1 1) + k (n n 1 )(n + n 1 1) n n 1 + 1 = (n 1 1) n n 1 + 1 ( 1 n ) n 1 (n + n 1 1) + (n 1)(n n 1 + 1) n 1 + > 0. n n 1 + 1 n n 1 + 1 Thus, η(k k + (K n1 K n )) > λ 0 and η(k k + (K n1 1 K n+1)) > λ 0. Since h(λ) is a decreasing function of λ, g n1,n (λ) g n1 1,n +1(λ) < 0 for λ > λ 0, which yields η(k k +(K n1 K n )) > η(k k +(K n1 1 K n+1)).

Volume 3, pp. 438-446, December 017. 443 On the Distance Signless Laplacian Spectral Radius of Graphs and Digraphs Lemma 3.4. [11] Let G be a connected graph with u, v V (G) and uv / E(G). Then η(g) > η(g+uv). Theorem 3.5. Among all the connected graphs of order n with connectivity at most k and minimum degree δ, G k,δ,n uniquely minimizes the distance signless Laplacian spectral radius. Proof. The case k = n 1 is trivial, and so we assume that k n. Let G be a graph in G k,n with the minimum distance signless Laplacian spectral radius, and S be a k-vertex cut of G. Now we verify the following two claims. Claim 1. G S consists exactly of two components. Suppose to the contrary that G S contains G 1, G and G 3. Then there must be u G and v G 3 such that uv E(G). Thus, by Lemma 3.4, η(g) > η(g + uv) while G + uv G k,n. This contradiction shows that G S = G 1 G. Claim. G i and G[V (G i ) S] (i = 1, ) are all complete graphs. If there is uv / E(G i ), then G+uv G k,n. By Lemma 3.4, η(g) > η(g+uv), a contradiction. Similarly, G[V (G i ) S] is also a complete graphs. By the above argument we can determine that G = K k +(K n1 K n ). Since δ k, we have n i δ k+1. If n 1 δ k +, without loss of generality, we suppose n n 1, then η(g) > η(k k + (K n1 1 K n+1)) while K k + (K n1 1 K n+1) G k,n by Lemma 3.3. This contradiction shows that n 1 = δ k + 1 and n = n δ 1, that is, G = G k,δ,n. Let C(n, k) be the set of connected graphs with n vertices and connectivity k, where 1 k n, then the following result can be obtained immediately. Corollary 3.6. [1] Let G C(n, k),where 1 k n. Then, η(g) η(k k + (K 1 K n k 1 )) with equality if and only if G = K k + (K 1 K n k 1 ). 4. The minimal distance signless Laplacian spectral radius for digraphs with given vertex connectivity. Similar to undirected graph, we use D( G) = (d ij ) n n and D Q ( G) = Diag(T r) + D( G) to denote the distance matrix and distance signless Laplacian matrix of G, respectively, where d ij = d G (v i, v j ) denote the length of shortest dipath from v i to v j in G and Diag(T r) is the diagonal matrix with D i. Clearly, the matrix D Q ( G) is irreducible when G is strongly connected. The eigenvalue of D Q ( G) with the largest modulus is the distance signless Laplacian spectral radius of G, denoted by η( G). For a strongly connected digraph G = (V ( G), E( G)), a set of vertices S V ( G) is a vertex cut if G S is not strongly connected. The vertex connectivity of G, denoted by κ( G), is the minimum number of vertices whose deletion yields the resulting digraph non-strongly connected. In this section, we characterize the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity. Let D n,k denote the set of strongly connected digraphs with order n and vertex connectivity κ( G) = k. If k = n 1, then D n,n 1 = { K n }. Thus, we only need to discuss the cases 1 k n. Let G 1 G denote the digraph obtained from two disjoint digraphs G 1, G with vertex set V ( G 1 ) V ( G ) and arc set E = E( G 1 ) E( G ) {(u, v), (v, u) u V ( G 1 ), v V ( G )}. Let 1 κ n, and K k,s n s denote the digraphs K k ( K s K n k s ) E 1, where E 1 = {(u, v) u V ( K s ), v V ( K n k s )}. Let K n,k = { K k,s n s 1 s n s 1}. A digraph G is a minimizing digraph of D n,k if G D n,k and η( G) = min{η( G) G D n,k }.

Volume 3, pp. 438-446, December 017. Dan Li, Guoping Wang, and Jixiang Meng 444 Let J a b be the a b matrix whose entries are all equal to 1, I n be the n n unit matrix and A( G) be the arc set of digraph G. Lemma 4.1. [] Let G be an arbitrary strongly connected digraph with vertex connectivity k. Suppose that S is a k-vertex cut of G and G 1,..., G s are the strongly connected components of G S. Then there exists an ordering of G 1,..., G s such that, for 1 i s and v V ( G i ), every tail of v in G 1,..., G i. Lemma 4.. [8] Let G be a strongly connected digraph with u, v V ( G) and uv / A( G). η( G) > η( G + uv). By Lemma 4.1, we know that there exists a strongly connected component of G S, say G 1, such that no vertex of V ( G 1 ) has in-neighbors in G S G 1. Let G 1 = G S G 1. Next, we construct a new digraph H by adding to G any legal arcs from G[V ( G 1 )] to G[V ( G 1 )] G[V ( G 1 )] or any legal arcs from G[V ( G1 )] to G[V ( G 1 )] that were not present in G. Obviously, H is also k-connected and H K k,s n s. Therefore, by Lemma 4., we have the digraphs which achieve the minimum distance signless Laplacian spectral radius among all digraphs in D n,k must be in K n,k. Theorem 4.3. The digraph K k,1 n 1 is the minimizing digraph among all digraphs in K k,s n s. Furthermore, if G D n,k, then η( G) 3n + 1+ n +6n 8k 7 with equality if and only if G = K k,1 n 1. Proof. Let G be an arbitrary digraph in K k,s n s and S be a k-vertex cut of G. Suppose that G 1, G (with V ( G 1 ) = n 1 and V ( G 1 ) = n = n k n 1, respectively) are two strongly connected components of G S and with arcs E = {u 1 u E u 1 V ( G 1 ), u V ( G )}. Clearly, 1 n 1 n k 1. Let D Q ( G) be the distance signless Laplacian matrix of G. Then D Q ( G) = J n1 n 1 + (n )I n1 J n1 k J n1 n J k n1 J k k + (n )I k J k n J n n 1 J n k J n n + (n + n 1 )I n1 Then and where f(λ) = Thus, P DQ ( G) = λi n D Q ( G) = (λ n + ) n1+k (λ n n 1 + ) n 1 f(λ), λ n n 1 + k n n 1 λ n k + n n 1 k λ n n 1 n +. P DQ ( = (λ n + G) )n1+k 1 (λ n n 1 + ) n 1 [λ + ( k n n 1 n + 4)λ + n(k + n 1 + n ) + (k + n n )n 1 + n + n 1 4n k 4n 1 n + 4] = (λ n + ) n1+k 1 (λ n n 1 + ) n 1 [λ + ( 3n n 1 + 4)λ + kn 1 + n + n 1 6n n 1 + 4]. Obviously, and 3n + n 1 + (k + n ) + 8n 1 n > n 3n + n 1 + (k + n ) + 8n 1 n > n + n 1.

Volume 3, pp. 438-446, December 017. 445 On the Distance Signless Laplacian Spectral Radius of Graphs and Digraphs Thus, η( G) = 3n + n 1 + (k + n ) + 8n 1 n. Since n = n k n 1, we can assume that and (f(n 1 )) (n 1 ) η( G) = f(n 1 ) = 3n + n 1 + 7n 1 + (6n 8k)n 1 + n ( 8k + 6n 14n 1 ) 7 = 8( 8kn 1 + n + 6nn 1 7n 1 ) 3 8kn 1 + n + 6nn 1 7n 1 < 0. For fixed n and k, we can assume that n > k+ since in case n = k+ there is only one value of n 1 under consideration. Since (f(n 1)) (n 1) < 0, the function f(n 1 ) is a convex function with respect to n 1. Combining f(n 1 ) > 0 for 1 n 1 n k 1, we know that the minimal value of η( G) must be taken at either n 1 = 1 or at n 1 = n k 1. Let g(x) = 7x + (6n 8k)x + n, then β = g(1) = (n + 7)(n 1) 8k > 0 and α = g(n k 1) = (k 3) + 8(n ) > 0. Then f(n k 1) f(1) = n k + α β = 1 ( (n k ) 1 n + k ). α + β If f(n k 1) f(1) 0, then we can get that { α + β n + k, α β n + k +. By the above inequalities, we have α (k + 1) which reduces to n k + 1, a contradiction. Thus, f(n k 1) f(1) > 0, i.e., η( K k,n k 1 k+1 ) > η( K k,1 n 1 ) and η( G) η( K k,1 n 1 ) = 3n + 1+ n +6n 8k 7 with equality if and only if G = K k,1 n 1. Acknowledgment. The authors are grateful to the anonymous referees for their valuable comments and helpful suggestions, which have considerably improved the presentation of this paper. This work was supported by NSFC (no. 11531011 and no. 11461071) and NSFXJ (no. 015KL019). REFERENCES [1] M. Aouchiche and P. Hansen. Two Laplacians for the distance matrix of a graph. Linear Algebra and its Applications, 439:1 33, 013. [] J. Bondy and U. Murty. Graph Theory with Applications. Macmillan, London, 1976. [3] S.S. Bose, M. Nath, and S. Paul. On the maximal distance spectral radius of graphs without a pendent vertex. Linear Algebra and its Applications, 438:460 478, 013. [4] Y. Chen, H. Lin, and J. Shu. Sharp upper bounds on the distance spectral radius of a graph. Linear Algebra and its Applications, 439:659 666, 013. [5] W. Hong and L. You. Some sharp bounds on the distance signless Laplacian spectral radius of graphs. Mathematics, 013. [6] A. Ilić. Distance spetral radius of trees with given matching number. Discrete Applied Mathematics, 158:1799 1806, 010. [7] G. Indulal. Sharp bounds on the distance spectral radius and the distance energy of graphs. Linear Algebra and its Applications, 430:106 113, 009. [8] D. Li, G.P. Wang, and J.X. Meng. Some results on the distance and distance signless Laplacian spectral radius of graphs and digraphs. Applied Mathematics and Computation, 93:18 5, 017.

Volume 3, pp. 438-446, December 017. Dan Li, Guoping Wang, and Jixiang Meng 446 [9] H. Lin and X. Lu. Bounds on the distance signless laplacian spectral radius in terms of clique number. Linear Multilinear Algebra, 63(9):1750 1759, 015. [10] H. Lin, W. Yang, H. Zhang, and J. Shu. The distance spectral radius of digraphs with given connectivity. Discrete Mathematics, 31:1849 1856, 01. [11] R. Xing, B. Zhou, and J. Li. On the distance signless laplacian spectral radius of graphs. Linear Multilinear Algebra, 6(10):1377 1387, 014. [1] R. Xing and B. Zhou. On the distance and distance signless Laplacian spectral radii of bicyclic graphs. Linear Algebra and its Applications, 439:3955 3963, 013.