Bounds for Kirchhoff index and Laplacian-energy-like invariant of some derived graphs of a regular graph

Abstract

The Kirchhoff index and Laplacian-energy-like invariant of a connected graph [Formula: see text], denoted by [Formula: see text] and [Formula: see text], are given by the number of vertex times the sum of the reciprocals of all nonzero Laplacian eigenvalues of [Formula: see text] and the sum of the square roots of all Laplacian eigenvalues of [Formula: see text], respectively. In this paper, we have obtained the Laplacian eigenvalues of some derived graphs, such as double graph, extended double cover and Mycielskian of an [Formula: see text]-regular graph [Formula: see text], in terms of the adjacency eigenvalues of [Formula: see text] and hence, we obtain some upper bounds of Kirchhoff index and Laplacian-energy-like (LEL) invariant of those derived graphs in terms of [Formula: see text], number of vertices and algebraic connectivity of [Formula: see text]. We have shown that the bounds obtained here are better than some existing bounds. We have also obtained the exact formulae for Kirchhoff index and LEL invariant of those derived graph when [Formula: see text] is a complete graph or a complete bipartite graph.