This paper studies the problem of the spectral radius of the uniform hypergraph determined by the signless Laplacian matrix.The upper bound of the spectral radius of a uniform hypergraph is obtained by using Rayleigh ...This paper studies the problem of the spectral radius of the uniform hypergraph determined by the signless Laplacian matrix.The upper bound of the spectral radius of a uniform hypergraph is obtained by using Rayleigh principle and the perturbation of the spectral radius under moving the edge operation,and the extremal hypergraphs are characterized for both supertree and unicyclic hypergraphs.The spectral radius of the graph is generalized.展开更多
Abstract For a simple undirected graph G with fixed size m≥2k(k∈Z^(+))and maximum degree Δ(G)≤m-k,we give an upper bound on the signless Laplacian spectral radius q(G)of G.For two connected graphs G_(1) and G_(2) ...Abstract For a simple undirected graph G with fixed size m≥2k(k∈Z^(+))and maximum degree Δ(G)≤m-k,we give an upper bound on the signless Laplacian spectral radius q(G)of G.For two connected graphs G_(1) and G_(2) with size m≥8,employing this upper bound,we prove that q(G_(1))>q(G_(2))if Δ(G_(1))>Δ(G_(2))+1 and Δ(G_(1))≥m/2+2.For triangle-free graphs,we prove two stronger results.As an application,we completely characterize the graph with maximal signless Laplacian spectral radius among all graphs with size m and circumference c for m≥max{2c,c+9},which partially answers the question proposed by Chen et al.in[Linear Algebra Appl.,2022,645:123–136].展开更多
For a simple graph G,let matrix Q(G)=D(G) + A(G) be it's signless Laplacian matrix and Q G (λ)=det(λI Q) it's signless Laplacian characteristic polynomial,where D(G) denotes the diagonal matrix of vertex deg...For a simple graph G,let matrix Q(G)=D(G) + A(G) be it's signless Laplacian matrix and Q G (λ)=det(λI Q) it's signless Laplacian characteristic polynomial,where D(G) denotes the diagonal matrix of vertex degrees of G,A(G) denotes its adjacency matrix of G.If all eigenvalues of Q G (λ) are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K(n_1,n_2,···,n_t).We prove that the complete t-partite graphs K(n,n,···,n)t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K(m,···,m)s(n,···,n)t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K(m,···,m,)s1(n,···,n,)s2(l,···,l)s3.展开更多
Let G be a connected graph on n vertices with chromatic number k, and let ρ（G） be the distance signless Laplacian spectral radius of G. We show that ρ（G） ≥ 2n ＋ 2[n/k] - 4, with equality if and only if G is a...Let G be a connected graph on n vertices with chromatic number k, and let ρ（G） be the distance signless Laplacian spectral radius of G. We show that ρ（G） ≥ 2n ＋ 2[n/k] - 4, with equality if and only if G is a regular Turan graph.展开更多
Let G be a simple graph with n vertices and m edges. In this paper, we present some new upper bounds for the adjacency and the signless Laplacian spectral radius of graphs in which every pair of adjacent vertices has ...Let G be a simple graph with n vertices and m edges. In this paper, we present some new upper bounds for the adjacency and the signless Laplacian spectral radius of graphs in which every pair of adjacent vertices has at least one common adjacent vertex. Our results improve some known upper bounds. The main tool we use here is the Lagrange identity.展开更多
Let G be a simple graph and let Q（G） be the signless Laplacian matrix of G. In this paper we obtain some results on the spectral perturbation of the matrix Q（G） under an edge addition or an edge contraction.
Let G be a simple graph. We first show that δ≥di-√[i/2][i/2], where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph...Let G be a simple graph. We first show that δ≥di-√[i/2][i/2], where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from G. In addition, for the distance signless Laplacian spectral radius ρQ（G）, we determine the extremal graphs with the minimum ρQ（G） among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.展开更多
A tricyclic graph G =（V（G）, E（G）） is a connected and simple graph such that｜E（G）｜ = ｜V（G）｜＋2. Let Tg nbe the set of all tricyclic graphs on n vertices with girth g. In this paper, we will show that ther...A tricyclic graph G =（V（G）, E（G）） is a connected and simple graph such that｜E（G）｜ = ｜V（G）｜＋2. Let Tg nbe the set of all tricyclic graphs on n vertices with girth g. In this paper, we will show that there exists the unique graph which has the largest signless Laplacian spectral radius among all tricyclic graphs with girth g containing exactly three（resp., four）cycles. And at the same time, we also give an upper bound of the signless Laplacian spectral radius and the extremal graph having the largest signless Laplacian spectral radius in Tg n,where g is even.展开更多
Let G be a simple connected graph with pendant vertex set ？V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q（G）. The signless Dirichlet eigenvalue is a real number λ such that the...Let G be a simple connected graph with pendant vertex set ？V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q（G）. The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ≠ 0 on V（G） such that Q（G）f（u） = λf（u） for u ∈ V_0 and f（u） = 0 for u ∈ ？V. The signless Dirichlet spectral radiusλ（G） is the largest signless Dirichlet eigenvalue. In this paper, the unicyclic graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with a given degree sequence are characterized.展开更多
The signless Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the smallest eigenvalue of its signless Laplacian matrix. In this paper, we determine the first to llth large...The signless Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the smallest eigenvalue of its signless Laplacian matrix. In this paper, we determine the first to llth largest signless Laplacian spectral radii in the class of bicyclic graphs with n vertices. Moreover, the unique bicyclic graph with the largest or the second largest signless Laplacian spread among the class of connected bicyclic graphs of order n is determined, respectively.展开更多
Let G be a digraph and A（G） be the adjacency matrix of G. Let D（G） be the diagonal matrix with outdegrees of vertices of G and Q（G） = D（G） ＋ A（G） be the signless Laplacian matrix of G. The spectral radius o...Let G be a digraph and A（G） be the adjacency matrix of G. Let D（G） be the diagonal matrix with outdegrees of vertices of G and Q（G） = D（G） ＋ A（G） be the signless Laplacian matrix of G. The spectral radius of Q（G） is called the signless Laplacian spectral radius of 9. In this paper, we determine the unique digraph which attains the maximum （or minimum） signless Laplacian spectral radius among all strongly connected bicyclic digraphs. Furthermore, we prove that any strongly connected bicyclic digraph is determined by the signless Laplacian spectrum.展开更多
In this paper, we found the bounds of the extreme eigenvalues of normalized Laplacian matrices and signless Laplacian matrices by using their traces. In addition, we found the bounds for k-th eigenvalues of normalized...In this paper, we found the bounds of the extreme eigenvalues of normalized Laplacian matrices and signless Laplacian matrices by using their traces. In addition, we found the bounds for k-th eigenvalues of normalized Laplacian matrix and signless Laplacian matrix.展开更多
In this paper, we determine the unique graph with the largest signless Laplacian spectral radius among all the tricyclic graphs with n vertices and k pendant vertices.
A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of ...A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C_4-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the(combinatorial)展开更多
This paper mainly researches on the signless laplacian spectral radius of bipartite graphs Dr(m1,m2;n1,n2). We consider how the signless laplacian spectral radius of Dr(m1,m2;n1,n2)?changes under some special cases. A...This paper mainly researches on the signless laplacian spectral radius of bipartite graphs Dr(m1,m2;n1,n2). We consider how the signless laplacian spectral radius of Dr(m1,m2;n1,n2)?changes under some special cases. As application, we give two upper bounds on the signless laplacian spectral radius of Dr(m1,m2;n1,n2), and determine the graphs that obtain the upper bounds.展开更多
A connected graph G=(V,E)is called a quasi-tree graph if there exists a vertex v0∈V(G)such that G-v0 is a tree.In this paper,we determine all quasi-tree graphs of order n with the second largest signless Laplacian ei...A connected graph G=(V,E)is called a quasi-tree graph if there exists a vertex v0∈V(G)such that G-v0 is a tree.In this paper,we determine all quasi-tree graphs of order n with the second largest signless Laplacian eigenvalue greater than or equal to n-3.As an application,we determine all quasi-tree graphs of order n with the sum of the two largest signless Laplacian eigenvalues greater than to 2 n-5/4.展开更多
基金Supported by Natural Science Foundation of HuBei Province(2022CFB299).
文摘This paper studies the problem of the spectral radius of the uniform hypergraph determined by the signless Laplacian matrix.The upper bound of the spectral radius of a uniform hypergraph is obtained by using Rayleigh principle and the perturbation of the spectral radius under moving the edge operation,and the extremal hypergraphs are characterized for both supertree and unicyclic hypergraphs.The spectral radius of the graph is generalized.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12071411,12171222)。
文摘Abstract For a simple undirected graph G with fixed size m≥2k(k∈Z^(+))and maximum degree Δ(G)≤m-k,we give an upper bound on the signless Laplacian spectral radius q(G)of G.For two connected graphs G_(1) and G_(2) with size m≥8,employing this upper bound,we prove that q(G_(1))>q(G_(2))if Δ(G_(1))>Δ(G_(2))+1 and Δ(G_(1))≥m/2+2.For triangle-free graphs,we prove two stronger results.As an application,we completely characterize the graph with maximal signless Laplacian spectral radius among all graphs with size m and circumference c for m≥max{2c,c+9},which partially answers the question proposed by Chen et al.in[Linear Algebra Appl.,2022,645:123–136].
基金Supported by the NSFC(60863006)Supported by the NCET(-06-0912)Supported by the Science-Technology Foundation for Middle-aged and Yong Scientist of Qinghai University(2011-QGY-8)
文摘For a simple graph G,let matrix Q(G)=D(G) + A(G) be it's signless Laplacian matrix and Q G (λ)=det(λI Q) it's signless Laplacian characteristic polynomial,where D(G) denotes the diagonal matrix of vertex degrees of G,A(G) denotes its adjacency matrix of G.If all eigenvalues of Q G (λ) are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K(n_1,n_2,···,n_t).We prove that the complete t-partite graphs K(n,n,···,n)t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K(m,···,m)s(n,···,n)t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K(m,···,m,)s1(n,···,n,)s2(l,···,l)s3.
基金Supported by National Natural Science Foundation of China(Grant Nos.1107100211371028)+4 种基金Program for New Century Excellent Talents in University(Grant No.NCET-10-0001)Key Project of Chinese Ministry of Education(Grant No.210091)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20103401110002)Natural Science Research Foundation of Department of Education of Anhui Province(Grant No.KJ2013A196)Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University(Grant No.KJJQ1001)
文摘Let G be a connected graph on n vertices with chromatic number k, and let ρ（G） be the distance signless Laplacian spectral radius of G. We show that ρ（G） ≥ 2n ＋ 2[n/k] - 4, with equality if and only if G is a regular Turan graph.
基金Supported by the National Natural Science Foundation of China(11471077)the Open Research Fund of Key Laboratory of Spatial Data Mining and Information Sharing of MOE(2018LSDMIS09)Foundation of Key Laboratory of Intelligent Metro of Universities in Fujian Province(53001703)
文摘Let G be a simple graph with n vertices and m edges. In this paper, we present some new upper bounds for the adjacency and the signless Laplacian spectral radius of graphs in which every pair of adjacent vertices has at least one common adjacent vertex. Our results improve some known upper bounds. The main tool we use here is the Lagrange identity.
基金Supported by the National Natural Science Foundation of China(11071002)the Anhui Natural ScienceFoundation of China(11040606M14)NSF of Department of Education of Anhui Province(KJ2011A195)
文摘Let G be a simple graph and let Q（G） be the signless Laplacian matrix of G. In this paper we obtain some results on the spectral perturbation of the matrix Q（G） under an edge addition or an edge contraction.
基金Supported by the National Natural Science Foundation of China(Grant No.11171343)
文摘Let G be a simple graph. We first show that δ≥di-√[i/2][i/2], where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from G. In addition, for the distance signless Laplacian spectral radius ρQ（G）, we determine the extremal graphs with the minimum ρQ（G） among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.
基金Supported by the National Natural Science Foundation of China(Grant No.11171273)
文摘A tricyclic graph G =（V（G）, E（G）） is a connected and simple graph such that｜E（G）｜ = ｜V（G）｜＋2. Let Tg nbe the set of all tricyclic graphs on n vertices with girth g. In this paper, we will show that there exists the unique graph which has the largest signless Laplacian spectral radius among all tricyclic graphs with girth g containing exactly three（resp., four）cycles. And at the same time, we also give an upper bound of the signless Laplacian spectral radius and the extremal graph having the largest signless Laplacian spectral radius in Tg n,where g is even.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1127125611601208)
文摘Let G be a simple connected graph with pendant vertex set ？V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q（G）. The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ≠ 0 on V（G） such that Q（G）f（u） = λf（u） for u ∈ V_0 and f（u） = 0 for u ∈ ？V. The signless Dirichlet spectral radiusλ（G） is the largest signless Dirichlet eigenvalue. In this paper, the unicyclic graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with a given degree sequence are characterized.
基金Supported by the National Natural Science Foundation of China(Grant No.11171273)Graduate Starting Seed Fund of Northwestern Polytechnical University(Grant No.Z2014173)
文摘The signless Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the smallest eigenvalue of its signless Laplacian matrix. In this paper, we determine the first to llth largest signless Laplacian spectral radii in the class of bicyclic graphs with n vertices. Moreover, the unique bicyclic graph with the largest or the second largest signless Laplacian spread among the class of connected bicyclic graphs of order n is determined, respectively.
基金Supported by the National Natural Science Foundation of China(Grant No.11171273)
文摘Let G be a digraph and A（G） be the adjacency matrix of G. Let D（G） be the diagonal matrix with outdegrees of vertices of G and Q（G） = D（G） ＋ A（G） be the signless Laplacian matrix of G. The spectral radius of Q（G） is called the signless Laplacian spectral radius of 9. In this paper, we determine the unique digraph which attains the maximum （or minimum） signless Laplacian spectral radius among all strongly connected bicyclic digraphs. Furthermore, we prove that any strongly connected bicyclic digraph is determined by the signless Laplacian spectrum.
文摘In this paper, we found the bounds of the extreme eigenvalues of normalized Laplacian matrices and signless Laplacian matrices by using their traces. In addition, we found the bounds for k-th eigenvalues of normalized Laplacian matrix and signless Laplacian matrix.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1087120461170311)the Fundamental Research Funds for the Central Universities(Grant No.09CX04003A)
文摘In this paper, we determine the unique graph with the largest signless Laplacian spectral radius among all the tricyclic graphs with n vertices and k pendant vertices.
基金Supported by the National Natural Science Foundation of China（11171273） Supported by the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical Uni- versity（Z2016170）
文摘A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C_4-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the(combinatorial)
文摘This paper mainly researches on the signless laplacian spectral radius of bipartite graphs Dr(m1,m2;n1,n2). We consider how the signless laplacian spectral radius of Dr(m1,m2;n1,n2)?changes under some special cases. As application, we give two upper bounds on the signless laplacian spectral radius of Dr(m1,m2;n1,n2), and determine the graphs that obtain the upper bounds.
基金Supported by the National Natural Science Foundation of China(Grant No.11771443)the Fundamental Research Funds for the Central Universities(Grant No.2018BSCXB24)the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX18JL980).
文摘A connected graph G=(V,E)is called a quasi-tree graph if there exists a vertex v0∈V(G)such that G-v0 is a tree.In this paper,we determine all quasi-tree graphs of order n with the second largest signless Laplacian eigenvalue greater than or equal to n-3.As an application,we determine all quasi-tree graphs of order n with the sum of the two largest signless Laplacian eigenvalues greater than to 2 n-5/4.
