In this paper,the definition of absolutely balanced and uniformly balanced for graphs are introduced,the difference between balance graphs are pointed out.Using（p,p＋1）-graph as an example,we explained the existence...In this paper,the definition of absolutely balanced and uniformly balanced for graphs are introduced,the difference between balance graphs are pointed out.Using（p,p＋1）-graph as an example,we explained the existence of this difference and obtained some new results.展开更多
Because of cloud computing's high degree of polymerization calculation mode, it can't give full play to the resources of the edge device such as computing, storage, etc. Fog computing can improve the resource ...Because of cloud computing's high degree of polymerization calculation mode, it can't give full play to the resources of the edge device such as computing, storage, etc. Fog computing can improve the resource utilization efficiency of the edge device, and solve the problem about service computing of the delay-sensitive applications. This paper researches on the framework of the fog computing, and adopts Cloud Atomization Technology to turn physical nodes in different levels into virtual machine nodes. On this basis, this paper uses the graph partitioning theory to build the fog computing's load balancing algorithm based on dynamic graph partitioning. The simulation results show that the framework of the fog computing after Cloud Atomization can build the system network flexibly, and dynamic load balancing mechanism can effectively configure system resources as well as reducing the consumption of node migration brought by system changes.展开更多
Let G =（V1,V2,E） be a balanced bipartite graph with2 n vertices.The bipartite binding number of G,denoted by B（G）,is defined to be n if G =Kn and min i∈{1,2}｜N（S）｜〈n min ｜N（S）｜/｜S｜otherwise.We call G b...Let G =（V1,V2,E） be a balanced bipartite graph with2 n vertices.The bipartite binding number of G,denoted by B（G）,is defined to be n if G =Kn and min i∈{1,2}｜N（S）｜〈n min ｜N（S）｜/｜S｜otherwise.We call G bipancyclic if it contains a cycle of every even length m for 4 ≤ m ≤ 2n.A theorem showed that if G is a balanced bipartite graph with 2n vertices,B（G） 〉 3 / 2 and n 139,then G is bipancyclic.This paper generalizes the conclusion as follows：Let 0 〈 c 〈 3 / 2 and G be a 2-colmected balanced bipartite graph with 2n（n is large enough） vertices such that B（G） c and δ（G）（2-c）n/（3-c）＋2/3.Then G is bipancyclic.展开更多
A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Lapl...A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.展开更多
In this paper, we obtain the following result: Let k, n 1 and n 2 be three positive integers, and let G = (V 1,V 2;E) be a bipartite graph with |V1| = n 1 and |V 2| = n 2 such that n 1 ? 2k + 1, n 2 ? 2k + 1 and |n 1 ...In this paper, we obtain the following result: Let k, n 1 and n 2 be three positive integers, and let G = (V 1,V 2;E) be a bipartite graph with |V1| = n 1 and |V 2| = n 2 such that n 1 ? 2k + 1, n 2 ? 2k + 1 and |n 1 ? n 2| ? 1. If d(x) + d(y) ? 2k + 2 for every x ∈ V 1 and y ∈ V 2 with xy $ \notin $ E(G), then G contains k independent cycles. This result is a response to Enomoto’s problems on independent cycles in a bipartite graph.展开更多
In this paper, the problem of consensus for continuous time singular systems of multi-agent networks is considered. The definition of r-consensus is introduced for singular systems of multi-agent networks. Firstly, li...In this paper, the problem of consensus for continuous time singular systems of multi-agent networks is considered. The definition of r-consensus is introduced for singular systems of multi-agent networks. Firstly, linear systems with algebraic constraints are considered, and the corresponding results about consensus and average-consensus are derived. Then r-consensus and consensus problems of singular systems are investigated. Sufficient conditions of r-consensus and consensus are obtained,respectively. Finally, an illustrative example is given to show the effectiveness of the proposed method.展开更多
基金Supported by the Natural Science Foundation of Shanxi Province（2008011010） Supported by the Scientific Research and Key Subject Foundation of University of Science and Technology of Suzhou
文摘In this paper,the definition of absolutely balanced and uniformly balanced for graphs are introduced,the difference between balance graphs are pointed out.Using（p,p＋1）-graph as an example,we explained the existence of this difference and obtained some new results.
基金supported in part by the National Science and technology support program of P.R.China(No.2014BAH29F05)
文摘Because of cloud computing's high degree of polymerization calculation mode, it can't give full play to the resources of the edge device such as computing, storage, etc. Fog computing can improve the resource utilization efficiency of the edge device, and solve the problem about service computing of the delay-sensitive applications. This paper researches on the framework of the fog computing, and adopts Cloud Atomization Technology to turn physical nodes in different levels into virtual machine nodes. On this basis, this paper uses the graph partitioning theory to build the fog computing's load balancing algorithm based on dynamic graph partitioning. The simulation results show that the framework of the fog computing after Cloud Atomization can build the system network flexibly, and dynamic load balancing mechanism can effectively configure system resources as well as reducing the consumption of node migration brought by system changes.
基金Supported by the Scientific Research Fund of Hubei Provincial Education Department(B2015021)
文摘Let G =（V1,V2,E） be a balanced bipartite graph with2 n vertices.The bipartite binding number of G,denoted by B（G）,is defined to be n if G =Kn and min i∈{1,2}｜N（S）｜〈n min ｜N（S）｜/｜S｜otherwise.We call G bipancyclic if it contains a cycle of every even length m for 4 ≤ m ≤ 2n.A theorem showed that if G is a balanced bipartite graph with 2n vertices,B（G） 〉 3 / 2 and n 139,then G is bipancyclic.This paper generalizes the conclusion as follows：Let 0 〈 c 〈 3 / 2 and G be a 2-colmected balanced bipartite graph with 2n（n is large enough） vertices such that B（G） c and δ（G）（2-c）n/（3-c）＋2/3.Then G is bipancyclic.
基金supported by the NSF of China(No.19971056)SRP(No.03B019) from the Education Committee of Hunan Province
文摘A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.
基金supported by the Foundation for the Distinguished Young Scholars of Shandong Province (Grant No.2007BS01021)the Taishan Scholar Fund from Shandong Province,SRF for ROCS,SEMNational Natural Science Foundation of China (Grant No.60673047)
文摘In this paper, we obtain the following result: Let k, n 1 and n 2 be three positive integers, and let G = (V 1,V 2;E) be a bipartite graph with |V1| = n 1 and |V 2| = n 2 such that n 1 ? 2k + 1, n 2 ? 2k + 1 and |n 1 ? n 2| ? 1. If d(x) + d(y) ? 2k + 2 for every x ∈ V 1 and y ∈ V 2 with xy $ \notin $ E(G), then G contains k independent cycles. This result is a response to Enomoto’s problems on independent cycles in a bipartite graph.
基金supported by the National Natural Science Foundation of China under Grant Nos.61174141 and 61374025Research Awards Young and Middle-Aged Scientists of Shandong Province under Grant Nos.BS2011SF009 and BS2011DX019Excellent Youth Foundation of Shandong Province under Grant No.JQ201219
文摘In this paper, the problem of consensus for continuous time singular systems of multi-agent networks is considered. The definition of r-consensus is introduced for singular systems of multi-agent networks. Firstly, linear systems with algebraic constraints are considered, and the corresponding results about consensus and average-consensus are derived. Then r-consensus and consensus problems of singular systems are investigated. Sufficient conditions of r-consensus and consensus are obtained,respectively. Finally, an illustrative example is given to show the effectiveness of the proposed method.
