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

摘要

The Kirchhoff index and Laplacian-energy-like invariant of a connected graph G, denoted by Kf(G) and LEL(G), are given by the number of vertex times the sum of the reciprocals of all nonzero Laplacian eigenvalues of G and the sum of the square roots of all Laplacian eigenvalues of G, 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 r-regular graph G, in terms of the adjacency eigenvalues of C and hence, we obtain some upper bounds of Kirchhoff index and Laplacian-energy-like (LEL) invariant of those derived graphs in terms of r, number of vertices and algebraic connectivity of G. 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 G is a complete graph or a complete bipartite graph.