The detection of error and its correction is an important area of mathematics that is vastly constructed in all communication systems.Furthermore,combinatorial design theory has several applications like detecting or ...The detection of error and its correction is an important area of mathematics that is vastly constructed in all communication systems.Furthermore,combinatorial design theory has several applications like detecting or correcting errors in communication systems.Network(graph)designs(GDs)are introduced as a generalization of the symmetric balanced incomplete block designs(BIBDs)that are utilized directly in the above mentioned application.The networks(graphs)have been represented by vectors whose entries are the labels of the vertices related to the lengths of edges linked to it.Here,a general method is proposed and applied to construct new networks designs.This method of networks representation has simplified the method of constructing the network designs.In this paper,a novel representation of networks is introduced and used as a technique of constructing the group generated network designs of the complete bipartite networks and certain circulants.A technique of constructing the group generated network designs of the circulants is given with group generated graph designs(GDs)of certain circulants.In addition,the GDs are transformed into an incidence matrices,the rows and the columns of these matrices can be both viewed as a binary nonlinear code.A novel coding error detection and correction application is proposed and examined.展开更多
An edge-coloring of a graph G is an coloring of a graph G is an edge-coloring of G such assignment of colors to all the edges of G. A go- that each color appears at each vertex at least g(v) times. The maximum integ...An edge-coloring of a graph G is an coloring of a graph G is an edge-coloring of G such assignment of colors to all the edges of G. A go- that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a go-coloring with k colors is called the gc-chromatic index of G and denoted by X'gc (G). In this paper, we extend a result on edge-covering coloring of Zhang X'gc( ) = δg(G), and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy ' x'gc(G)=δg(G),where δg(G)=minv∈V(G){[d(v)/g(v)]}.展开更多
This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph th...This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result (Theorem 13), Grone and Merris' conjecture [The Laplacian spectrum of graph II. SIAM J. Discrete Math., 7, 221-229 (1994)] is partially proved.展开更多
In this paper we get some relations between α(G), α'(G), β(G), β'(G) and αT(G), βT(G). And all bounds in these relations are best possible, where α(G), α'(G),/3(G), β(G), αT(G) and ...In this paper we get some relations between α(G), α'(G), β(G), β'(G) and αT(G), βT(G). And all bounds in these relations are best possible, where α(G), α'(G),/3(G), β(G), αT(G) and βT(G) are the covering number, edge-covering number, independent number, edge-independent number (or matching number), total covering number and total independent number, respectively.展开更多
基金support from Taif University Researchers Supporting Project Number(TURSP-2020/031),Taif University,Taif,Saudi Arabia.
文摘The detection of error and its correction is an important area of mathematics that is vastly constructed in all communication systems.Furthermore,combinatorial design theory has several applications like detecting or correcting errors in communication systems.Network(graph)designs(GDs)are introduced as a generalization of the symmetric balanced incomplete block designs(BIBDs)that are utilized directly in the above mentioned application.The networks(graphs)have been represented by vectors whose entries are the labels of the vertices related to the lengths of edges linked to it.Here,a general method is proposed and applied to construct new networks designs.This method of networks representation has simplified the method of constructing the network designs.In this paper,a novel representation of networks is introduced and used as a technique of constructing the group generated network designs of the complete bipartite networks and certain circulants.A technique of constructing the group generated network designs of the circulants is given with group generated graph designs(GDs)of certain circulants.In addition,the GDs are transformed into an incidence matrices,the rows and the columns of these matrices can be both viewed as a binary nonlinear code.A novel coding error detection and correction application is proposed and examined.
基金Supported by Shandong Provincial Natural Science Foundation,China(Grant No.ZR2014JL001)the Shandong Province Higher Educational Science and Technology Program(Grant No.J13LI04)the Excellent Young Scholars Research Fund of Shandong Normal University of China
文摘An edge-coloring of a graph G is an coloring of a graph G is an edge-coloring of G such assignment of colors to all the edges of G. A go- that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a go-coloring with k colors is called the gc-chromatic index of G and denoted by X'gc (G). In this paper, we extend a result on edge-covering coloring of Zhang X'gc( ) = δg(G), and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy ' x'gc(G)=δg(G),where δg(G)=minv∈V(G){[d(v)/g(v)]}.
基金Supported by National Natural Science Foundation of China (Grant No. 10871204) and the Fundamental Research Funds for the Central Universities (Grant No. 09CX04003A)
文摘This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result (Theorem 13), Grone and Merris' conjecture [The Laplacian spectrum of graph II. SIAM J. Discrete Math., 7, 221-229 (1994)] is partially proved.
基金Supported by the National Natural Science Foundation of China (No. 10771091)Com2MaC-KOSEF (No.(E)ndzr09-15)
文摘In this paper we get some relations between α(G), α'(G), β(G), β'(G) and αT(G), βT(G). And all bounds in these relations are best possible, where α(G), α'(G),/3(G), β(G), αT(G) and βT(G) are the covering number, edge-covering number, independent number, edge-independent number (or matching number), total covering number and total independent number, respectively.