Maximum and minimum degree energies of p -splitting and p -shadow graphs

Abstract

Let v(i) and v(j) be two vertices of a graph G. The maximum degree matrix of G is given in [2] byd(ij) = {max{d(i), d(j)} if v(i) and v(j) are adjacent0 otherwise.Similarly the (i, j)-th entry of the minimum degree matrix is defined by taking the minimum degree instead of the maximum degree above, [1]. In this paper, we have elucidated a relation between maximum degree energy of p-shadow graphs with the maximum degree energy of its underlying graph. Similarly, a relation has been derived for minimum degree energy also. We disprove the results E-M(S'(G)) = 2E(M)(G) and E-m(S'(G)) = 2E(m)(G) given by Zheng-Qing Chu et al. [3] by giving some counterexamples.