The Definition of Bimean Graphs

Abstract

Let G be a(p,q) graph and let: V(G) \(\rightarrow\){0,1,2,…..,q} be an injection. The graph G is said to be a mean graph if for each edge there exists an induced map \(f^*:E(G)\rightarrow{1,2,…,q}\) defined by \(f^*(uv) = {f(u)+f(v) \over 2}\), if  \(f(u)+f(v)\) is even or \({f(u)+f(v)+1 \over 2}\), if \(f(u)+f(v)\) is odd. The line graph of G is a graph in which the vertices are the edges (lines) of G and the two points of L(G) are adjacent whenever the corresponding lines of G are adjacent. In this chapter, we investigate the meanness of both G and L(G) and thus introduce the definition of bimean graphs.