## Abstract

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).

Original language | English |
---|---|

Article number | e27756 |

Number of pages | 5 |

Journal | Heliyon |

Volume | 10 |

Issue number | 6 |

Early online date | 8 Mar 2024 |

DOIs | |

Publication status | Published - 30 Mar 2024 |

## Keywords

- Energy of a graph
- Graphs with self-loops
- Singular values