The spectrum of the edge corona of two graphs

Abstract

Given two graphs G1, with vertices 1,2,...,n and edges e1,e2,...,em, and G2, the edge corona G1 ¦ G2 of G1 and G2 is defined as the graph obtained by taking m copies of G2 and for each edge ek = ij of G, joining edges between the two end-vertices i,j of ek and each vertex of the k-copy of G2. In this paper, the adjacency spectrum and Laplacian spectrum of G1 ¦ G2 are given in terms of the spectrum and Laplacian spectrum of G1 and G2, respectively. As an application of these results, the number of spanning trees of the edge corona is also considered.