Strongly Regular Graphs And High-Effective Network's Construction

This article can be divided into two section: The first is the study on strongly regular graphs.And in the second section, we do some discussion on graph theory & group theory's application on networks, it means combinatorics & network & graph theory.A graph G is strongly regular graph with parameters(n,k,λ,u), whenever G is regular of degree k :Each pair of adjacent vertices hasλcommon neighbors , and each pair of non-adjacent vertices has u common neighbors. It is well known that the eigenvalues of a strongly regular graph with parameters (n,k,λ,u) , are k andθ,T which are the two roots of the quadratic equation x² -(λ-u)x-(k-u) = 0. The multiplicities m_θand m_T can be determined from the equations m_θ+m_T =n-1,k + m_θθ+ m_TT = 0.This paper is organized into three chapters , and the main results lie in chapter 2 . The first section gives the definition and simple charecteristics of strongly regular graph. In section 2 , i mention two classifications of strongly regular graph: conference graph & no conference graph . And do some research on their parameters properties , and provide necessary and suffcient & necessary condition . I find that both mK_r and its complement are imprimitive strongly regular graphs , and mK_r's complement is exactly complete m-partite graph K_m(r). Using properties of imprimitive strongly regular graph , I get the spectra of K_m(r) and find that the parameters of it and the graph K_m(r) can be uniquely determined mutually . Then is hyperenergetic strongly regular graphs . Using Igor Shparlinski's work on energy of some circulant graphs , I give one type of hyperenergetic strongly regular graphs with parameter ( 4n+1, 2n, n-1, n ) and provide all heperenergetic strongly regular graphs up to 25 vertices ( list in next section ) . And I find that K_m(r)'s similar graph denoted by K_m(r)^ is also heperenergetic . It can be written as tensor product of K_m and K_r. And I list all strongly regular graphs up to 25 vertices in table with study .Chapter 3 belongs to section two, in this section we focus on discussion of how to construct maximal (Δ,D)transitive graph, main for Cayley graph. And the Abelian group and the Semi-direct product group is the main construction group.