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

Let G be a graph on n vertices with vertex set $$ and let $$ with $$. Denote by $$, 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 $$ ($$) in terms of ordinary energy ($$), order (n) and number of self-loops (α). Recently, it is proved that for a bipartite graph $$, $$. Here we show that this inequality is strict for an unbalanced bipartite graph $$ with $$. In other words, we show that there exits no unbalanced bipartite graph $$ with $$ and $$.