On Aα-spectrum of joined union of graphs and its applications to power graphs of finite groups

Abstract

For a simple graph [Formula: see text], the generalized adjacency matrix [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the adjacency matrix and [Formula: see text] is the diagonal matrix of vertex degrees of [Formula: see text]. This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find the [Formula: see text]-spectrum of the joined union of graphs in terms of the spectrum of the adjacency matrices of its components and the zeros of the characteristic polynomials of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular graphs, the join of a regular graph with the union of two regular graphs of distinct degrees. As applications, we investigate the [Formula: see text]-spectrum of certain power graphs of finite groups.