Graph Eigenvalues Under a Graph Transformation

For a graph X and a digraph D, we define the beta transformation of X and the alpha transformation of D denoted by X-beta and D-alpha respectively. D-alpha is defined as the bipartite graph with vertex set V(D) x {0, 1} and edge set {{(v(i), 0), (v(j), 1)}vertical bar v(i)v(j) is an element of A(D)}. X-beta is defined as the bipartite graph with vertex set V(X) x {0, 1} and edge set {{(v(i), 0), (v(i), 1)}vertical bar v(i)v(j) is an element of A((X) over right arrow)} where (X) over right arrow is the associated digraph of X. In this paper, we give the relation between the eigenvalues of the digraph D and the graph D-alpha when the adjacency matrix of D is normal. Especially, we obtain the eigenvalues of D-alpha when D is some special Cayley digraph.