Locally Equienergetic Graphs

TL;DR

This paper investigates pairs of graphs that have identical local energy, defined by the change in graph energy upon vertex removal, exploring their properties and examples.

Contribution

It introduces the concept of locally equienergetic graphs and examines several pairs of such graphs, expanding understanding of graph energy variations.

Findings

Identified multiple pairs of locally equienergetic graphs.

Defined local energy at vertices based on energy difference after vertex removal.

Explored properties and examples of locally equienergetic graphs.

Abstract

For a given graph \( G \), let \( G^{(j)} \) denote the graph obtained by the deletion of vertex \( v_j \) from \( G \). The difference \( \mathscr{E}(G) - \mathscr{E}(G^{(j)}) \) quantifies the change in the energy of \( G \) upon the removal of \( v_j \), termed as the local energy of \( G \) at vertex $v_j$, as defined by Espinal and Rada in 2024. The local energy of $G$ at vertex $v$ is denoted by \(\mathscr{E}_G(v)\). The local energy of the graph \( G \), therefore, is the summation of these vertex-specific local energies across all vertices in \( V(G) \), expressed by \( e(G) = \sum \mathscr{E}_G(v) \). Two graphs of the same order are defined as locally equienergetic if they have identical local energy. In this paper, we have investigated several pairs of locally equienergetic graphs.