The Vertex Distance Complement Spectrum of Subdivision Vertex Join and Subdivision Edge Join of Two Regular Graphs

Abstract

The vertex distance complement (VDC) matrix C, of a connected graph G with vertex set consisting of n vertices, is a real symmetric matrix [cij] that takes the value n−dij where dij is the distance between the vertices vi and vj of G for i≠j and 0 otherwise. The vertex distance complement spectrum of the subdivision vertex join, G1⋁˙G2 and the subdivision edge join G1⋁−−G2 of regular graphs G1 and G2 in terms of the adjacency spectrum are determined in this paper.