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⊆V(G) with |S|=α. Denote by G_S, 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 (E(G_S)) in terms of ordinary energy (E(G)), order (n) and number of self-loops (α). Recently, it is proved that for a bipartite graph G_S, E(G_S)≥E(G). Here we show that this inequality is strict for an unbalanced bipartite graph G_S with 0<α<n. In other words, we show that there exits no unbalanced bipartite graph G_S with 0<α<n and E(G_S)=E(G).