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 simple connected graph with n vertices.The transmission Tv of a vertex v is defined to be the sum of the distances from v to all other vertices in G,that is,T_(v)=Σ_(u)∈Vd_(uv),where duv denotes the dista...Let G be a simple connected graph with n vertices.The transmission Tv of a vertex v is defined to be the sum of the distances from v to all other vertices in G,that is,T_(v)=Σ_(u)∈Vd_(uv),where duv denotes the distance between u and v.Let T_(1),…,T_(n)be the transmission sequence of G.Let D=(dij)_(n×n)be the distance matrix of G,and T be the transmission diagonal matrix diag(T_(1),…,T_(n)).The matrix Q(G)=T+D is called the distance signless Laplacian of G.In this paper,we provide the distance signless Laplacian spectrum of complete k-partite graph,and give some sharp lower and upper bounds on the distance signless Laplacian spectral radius q(G).展开更多
Complex networks have been a prominent topic of research for several years,spanning a wide range of fields from mathematics to computer science and also to social and biological sciences.The eigenvalues of the Seidel ...Complex networks have been a prominent topic of research for several years,spanning a wide range of fields from mathematics to computer science and also to social and biological sciences.The eigenvalues of the Seidel matrix,Seidel Signless Laplacian matrix,Seidel energy,Seidel Signless Laplacian energy,Maximum and Minimum energy,Degree Sum energy and Distance Degree energy of the Unitary Cayley graphs[UCG]have been calculated.Low-power devices must be able to transfer data across long distances with low delay and reliability.To overcome this drawback a small-world network depending on the unitary Cayley graph is proposed to decrease the delay and increase the reliability and is also used to create and analyze network communication.Small-world networks based on the Cayley graph have a basic construction and are highly adaptable.The simulation result shows that the small-world network based on unitary Cayley graphs has a shorter delay and is more reliable.Furthermore,the maximum delay is lowered by 40%.展开更多
Suppose that the vertex set of a graph G is V(G) ={v1,v2,...,vn}.The transmission Tr(vi) (or Di) of vertex vi is defined to be the sum of distances from vi to all other vertices.Let Tr(G) be the n × n diagonal ma...Suppose that the vertex set of a graph G is V(G) ={v1,v2,...,vn}.The transmission Tr(vi) (or Di) of vertex vi is defined to be the sum of distances from vi to all other vertices.Let Tr(G) be the n × n diagonal matrix with its (i,i)-entry equal to TrG(vi).The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G,defined as L(G) =Tr(G) + D(G),where D(G) is the distance matrix of G.In this paper,we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible.We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs.Moreover,we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees,and characterize extremal graphs.展开更多
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 the National Natural Science Foundation of China(Grant Nos.11801144,11701148)the Natural Science Foundation of Education Ministry of Henan Province(18B110005).
文摘Let G be a simple connected graph with n vertices.The transmission Tv of a vertex v is defined to be the sum of the distances from v to all other vertices in G,that is,T_(v)=Σ_(u)∈Vd_(uv),where duv denotes the distance between u and v.Let T_(1),…,T_(n)be the transmission sequence of G.Let D=(dij)_(n×n)be the distance matrix of G,and T be the transmission diagonal matrix diag(T_(1),…,T_(n)).The matrix Q(G)=T+D is called the distance signless Laplacian of G.In this paper,we provide the distance signless Laplacian spectrum of complete k-partite graph,and give some sharp lower and upper bounds on the distance signless Laplacian spectral radius q(G).
文摘Complex networks have been a prominent topic of research for several years,spanning a wide range of fields from mathematics to computer science and also to social and biological sciences.The eigenvalues of the Seidel matrix,Seidel Signless Laplacian matrix,Seidel energy,Seidel Signless Laplacian energy,Maximum and Minimum energy,Degree Sum energy and Distance Degree energy of the Unitary Cayley graphs[UCG]have been calculated.Low-power devices must be able to transfer data across long distances with low delay and reliability.To overcome this drawback a small-world network depending on the unitary Cayley graph is proposed to decrease the delay and increase the reliability and is also used to create and analyze network communication.Small-world networks based on the Cayley graph have a basic construction and are highly adaptable.The simulation result shows that the small-world network based on unitary Cayley graphs has a shorter delay and is more reliable.Furthermore,the maximum delay is lowered by 40%.
基金The authors are grateful to the two anonymous referees for their careful reading of this paper and strict criticisms, constructive corrections, and valuable comments on this paper, which have considerably improved the presentation of this paperThe first author was supported by the National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642+2 种基金The second author was supported by the National Natural Science Foundation of China (Grant No. 11771141)the Fundamental Research Fund for the Central Universities (No. 222201714049)The third author was supported by the National Natural Science Foundation of China (Grant No. 11371372).
文摘Suppose that the vertex set of a graph G is V(G) ={v1,v2,...,vn}.The transmission Tr(vi) (or Di) of vertex vi is defined to be the sum of distances from vi to all other vertices.Let Tr(G) be the n × n diagonal matrix with its (i,i)-entry equal to TrG(vi).The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G,defined as L(G) =Tr(G) + D(G),where D(G) is the distance matrix of G.In this paper,we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible.We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs.Moreover,we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees,and characterize extremal graphs.
基金Supported by National Natural Science Foundation of China(11071002)NFS of Anhui Province(11040606M14)NSF of Department of Education of Anhui Province(KJ2011A195,KJ2010B136)
基金Supported by the National Natural Science Foundation of China(No.11171273)the Natural Science Foundation of Shaanxi Province(No.SJ08A01)SRF for ROCS,SEM
基金Supported by the NSF of Department of Education of Anhui Province(KJ2011A195)the Innovation Fund for Graduates of Anhui Universitythe Anhui Provincial Natural Science Foundation(11040606M14)
