Energy and Vertex Energy of Modified Divisor Prime Graphs

TL;DR

This paper studies the energy and vertex energy of a modified divisor prime graph with self-loops, introducing a generalized vertex energy concept for such graphs.

Contribution

It defines and analyzes vertex energy for graphs with self-loops, specifically applied to the modified divisor prime graph.

Findings

Introduces a generalized vertex energy for graphs with self-loops.

Analyzes the energy of the modified divisor prime graph with a self-loop at vertex 1.

Provides mathematical properties of vertex energy in this context.

Abstract

This paper investigates the energy and vertex energy of the modified divisor prime graph $G^*_{Dp}(n)$, which is distinguished from the standard divisor prime graph by the inclusion of a self-loop at the vertex $1$. To facilitate this analysis, we introduce a generalized definition of vertex energy for graphs with self-loops and demonstrate its mathematical consistency.