Graph Eigenvalues Under a Graph Transformation

Abstract

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 \(V(D) \times \{0,1\}\) and edge set \(\{((v_i,0), (v_j, 1)) \mid v_i v_j \in A(D)\}\).\(X^\beta\) is defined as the bipartite graph with vertex set \(V(X) \times \{0,1\}\) and edge set \(\{((v_i,0), (v_j, 1)) \mid v_i v_j \in A(X)\}\), where \(X\) 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.