An equitable(O^(1)_(k),O^(2)_(k),...,O^(m)_(k))-partition of a graph G,which is also called a k cluster m-partition,is the partition of V(G)into m non-empty subsets V_(1),V_(2),...,Vm such that for every integer i in{...An equitable(O^(1)_(k),O^(2)_(k),...,O^(m)_(k))-partition of a graph G,which is also called a k cluster m-partition,is the partition of V(G)into m non-empty subsets V_(1),V_(2),...,Vm such that for every integer i in{1,2,...,m},G[Vi]is a graph with components of order at most k,and for each distinct pair i,j in{1,...,m},there is−1≤|Vi|−|Vj|≤1.In this paper,we proved that every planar graph G with minimum degreeδ(G)≥2 and girth g(G)≥12 admits an equitable(O_(1)^(7),O^(2)_(7),...,O^(m)_(7))-partition,for any integer m≥2.展开更多
Let G be a graph andαÎ[0,1),Nikiforov merged the adjacency matrix and the signless Laplacian matrix to A_(α)(G)=αD(G)+(1-α)A(G),where D(G)A(G)are the degree diagonal matrix and the adjacency matrix of G,respe...Let G be a graph andαÎ[0,1),Nikiforov merged the adjacency matrix and the signless Laplacian matrix to A_(α)(G)=αD(G)+(1-α)A(G),where D(G)A(G)are the degree diagonal matrix and the adjacency matrix of G,respectively.The spectral radius of A_(α)(G)is called byα-spectral radius of the graph G.In this paper,we study the perturbation of the complete multipartite graphsα-spectral radius when move a vertex from a part to other part of the complete multipartite graphs.Moreover,we give some conditions when theα-spectral Turán of graphs implies the Turán theorem of graphs.展开更多
The spectral radius of a graph is the maximum eigenvalues of its adjacency matrix. In this paper, using the property of quotient graph, the sharp upper bounds for the spectral radii of some adhesive graphs are determi...The spectral radius of a graph is the maximum eigenvalues of its adjacency matrix. In this paper, using the property of quotient graph, the sharp upper bounds for the spectral radii of some adhesive graphs are determined.展开更多
This paper investigates the controllability of general linear discrete-time multi-agent systems with directed and weighted signed networks by using graphic and algebraic methods.The nondelay and delay cases are consid...This paper investigates the controllability of general linear discrete-time multi-agent systems with directed and weighted signed networks by using graphic and algebraic methods.The nondelay and delay cases are considered respectively.For the case of no time delay,the upper bound condition of the controllable subspace is given by using the equitable partition method,and the influence of coefficient matrix selection of individual dynamics is illustrated.For the case of single delay and multiple delays,the equitable partition method is extended to deal with time-delay systems,and some conclusions are obtained.In particular,some simplified algebraic criteria for controllability of systems with time delay are obtained by using augmented system method and traditional algebraic controllability criteria.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1207126512271331)the Natural Science Foundation of Shandong Province(Grant No.ZR202102250232).
文摘An equitable(O^(1)_(k),O^(2)_(k),...,O^(m)_(k))-partition of a graph G,which is also called a k cluster m-partition,is the partition of V(G)into m non-empty subsets V_(1),V_(2),...,Vm such that for every integer i in{1,2,...,m},G[Vi]is a graph with components of order at most k,and for each distinct pair i,j in{1,...,m},there is−1≤|Vi|−|Vj|≤1.In this paper,we proved that every planar graph G with minimum degreeδ(G)≥2 and girth g(G)≥12 admits an equitable(O_(1)^(7),O^(2)_(7),...,O^(m)_(7))-partition,for any integer m≥2.
基金The National Science Foundation of China(12371349,12471331)。
文摘Let G be a graph andαÎ[0,1),Nikiforov merged the adjacency matrix and the signless Laplacian matrix to A_(α)(G)=αD(G)+(1-α)A(G),where D(G)A(G)are the degree diagonal matrix and the adjacency matrix of G,respectively.The spectral radius of A_(α)(G)is called byα-spectral radius of the graph G.In this paper,we study the perturbation of the complete multipartite graphsα-spectral radius when move a vertex from a part to other part of the complete multipartite graphs.Moreover,we give some conditions when theα-spectral Turán of graphs implies the Turán theorem of graphs.
文摘The spectral radius of a graph is the maximum eigenvalues of its adjacency matrix. In this paper, using the property of quotient graph, the sharp upper bounds for the spectral radii of some adhesive graphs are determined.
基金the National Natural Science Foundation of China under Grant Nos.61873136 and 62033007Taishan Scholars Climbing Program of Shandong Province of China and Taishan Scholars Project of Shandong Province of China under Grant No.ts20190930。
文摘This paper investigates the controllability of general linear discrete-time multi-agent systems with directed and weighted signed networks by using graphic and algebraic methods.The nondelay and delay cases are considered respectively.For the case of no time delay,the upper bound condition of the controllable subspace is given by using the equitable partition method,and the influence of coefficient matrix selection of individual dynamics is illustrated.For the case of single delay and multiple delays,the equitable partition method is extended to deal with time-delay systems,and some conclusions are obtained.In particular,some simplified algebraic criteria for controllability of systems with time delay are obtained by using augmented system method and traditional algebraic controllability criteria.
