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