Let be a graph on vertices with vertex set and let with . Denote by , the graph obtained from by adding a self-loop at each of the vertices in . In this note, we first give an upper bound and a lower bound for the energy of () in terms of ordinary energy (), order () and number of self-loops (). 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 .
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| http://dx.doi.org/10.1016/j.heliyon.2024.e27756 | DOI Listing |
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