Transmission delay of the network model based on the lexicographic product digraphs

Average distance of a digraph is an effective parameter to measure message delay of the network modeled by the digraph. In this paper, we use the lexicographic product method to construct a larger digraph from several specified small digraph. The digraph constructed by this way can contain the factor digraphs as subgraphs and preserve many desirable properties of the factor digraphs. By combining the theory of combination design and the way of algebra, we investigate the average distance parameters of the lexicographic product of digraphs, and giving a closed formula for the average distance of the network model based on the theory of the lexicographic product digraphs. This formula reveals that the network model based on the theory of the lexicographic product digraphs, which has a small transmission delay.

[1]  Wei Wang,et al.  Some results on the lexicographic product of vertex-transitive graphs , 2011, Appl. Math. Lett..

[2]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[3]  Donald E. Knuth,et al.  Stable Networks and Product Graphs , 1995 .

[4]  Peter Dankelmann.  Average Distance and Independence Number , 1994, Discret. Appl. Math..

[5]  Peter Winkler,et al.  Mean distance and minimum degree , 1997, J. Graph Theory.

[6]  Ján Plesník,et al.  On the sum of all distances in a graph or digraph , 1984, J. Graph Theory.

[7]  Junming Xu.  Topological Structure and Analysis of Interconnection Networks , 2002, Network Theory and Applications.

[8]  Michael L. Fredman,et al.  New Bounds on the Complexity of the Shortest Path Problem , 1976, SIAM J. Comput..

[9]  Zongben Xu,et al.  On the number of spanning trees of the lexicographic product of networks , 2012 .

[10]  V. Yegnanarayanan,et al.  On Some Graph Operations and Related Applications , 2009, Electron. Notes Discret. Math..

[11]  Feng Li,et al.  Vertex forwarding indices of the lexicographic product of graphs , 2014 .

[12]  Roger C. Entringer,et al.  Distance in graphs , 1976 .

[13]  F. Harary,et al.  A survey of the theory of hypercube graphs , 1988 .

[14]  Sheldon B. Akers,et al.  A Group-Theoretic Model for Symmetric Interconnection Networks , 1989, IEEE Trans. Computers.

[15]  Dong Liang,et al.  The number of spanning trees in a new lexicographic product of graphs , 2014, Science China Information Sciences.

[16]  Fan Chung Graham,et al.  The average distance and the independence number , 1988, Journal of Graph Theory.

[17]  Wei Wang,et al.  Graphs determined by their generalized characteristic polynomials , 2011 .