Automorphism Of The Graph L(K_m,n)

Let X be the simple digraph, whose vertex set, edge set and the full automorphism group is denoted by V(X),E(X) and Aut(X) respectively. Let G be a finite group and S be a subset not containing unit element of the G, we define the following Cayley digraph X=Cay(G,S) where V(X)=G,E(X)={(g,sg)│g∈G,s∈S}. In particular, if S=S-1,then X=Cay(G,S) is undirected and an undirected edge{u,v} is equivalent to two directed edges (u,v) and (v,u).We call a Cayley graph X=Cay(G,S) on a finite group G is normal if the right regular represent R(G) of G is normal in the full automorphism group Aut(X). In this paper, firstly, we introduce some basic concept, such as, isomorphism, automorphism, regular etc; Secondly,we introduces Cayley graph on the group and its automorphisms and prove some necessary lemmas; Finally,automorphism group of the line graph L(Km,n) of completely bipartite graph Km,n is discussed. Because the graph L(K1,n) is an isolated point, so we always assume that m and n are two positive integers with m=n.We first show that the graph L(Km,n) is a Cayley graph on the group Zm×Zn, and we obtain its full automorphism group. Furthermore, when m=n, the graph L(Km,n) is arc transitive. We also prove the graph L(Km,n), as the Cayley graph on the group Zm×Zn is not normal except for L(K1,2),L(K1,3),L(K2,2),L(K2,3) and L(K3,3).When max{m,n}>3, L(Km,n) is not formal.It is noteworthy that, when m=n,the graph L(Km,n) is a strongly regular graph with parameter (n2,1(n-1),n-2,2). That is, we give the full automorphism group of a class of strongly regular graph...