Second-stage spectrum of corona of two graphs

Abstract

The derived graph of a simple graph [Formula: see text], denoted by [Formula: see text], is the graph having the same vertex set as [Formula: see text], in which two vertices are adjacent if and only if their distance in [Formula: see text] is two. The spectrum of derived graph of [Formula: see text] is called the second-stage spectrum of [Formula: see text]. In this paper, we will determine the second-stage spectrum (Laplace and signless Laplace spectrum) of the corona of two regular graphs with diameter less than or equal to two. The energy of the derived graph is called the second-stage energy of [Formula: see text]. Here, we proved that the class of graphs [Formula: see text] is integral if and only if [Formula: see text] is a perfect square and the second-stage energy depends only on number of vertices. Moreover, we discuss some applications like the number of spanning trees, the Kirchhoff index and Laplacian-energy-like invariant of the newly constructed graph.