Relations between ordinary energy and energy of a self-loop graph

Let G be a graph on n vertices with vertex set V(G) and let S subset of V(G) with |S|=alpha. Denote by GS, the graph obtained from G by adding a self-loop at each of the vertices in S. In this note, we first give an upper bound and a lower bound for the energy of G(S) (epsilon(G(S))) in terms of ordinary energy (E(G)), order (n) and number of self-loops (alpha). Recently, it is proved that for a bipartite graph G(S), epsilon(G(S))>=epsilon(G). Here we show that this inequality is strict for an unbalanced bipartite graph G(S) with 0更多