Bounds on Path Energy of Graphs

TL;DR

This paper investigates upper bounds on the path energy $PE(M)$ of graphs by analyzing the spectrum of the path matrix $P(M)$. It derives concrete, parameter-based bounds such as $PE(M) \le 2(p-1)m$ and $PE(M) \le p(p-1)\Delta$ by establishing $|\beta| \le (p-1)\Delta$ for all path eigenvalues, and it relates $PE(M)$ to the classical graph energy via $E(M) \le \frac{p}{2} PE(M)$ (with equality for $M=K_2$). These bounds generalize previous results (e.g., $PE(M) \le 2(p-1)^2$) and connect path-energy to basic graph parameters (order $p$, size $m$, maximum degree $\Delta$) to quantify how path structure limits energy.

Abstract

Given a graph $M,$ path eigenvalues are eigenvalues of its path matrix. The path energy of a simple graph $M$ is equal to the sum of the absolute values of the path eigenvalues of the graph $M$ (Shikare et. al, 2018). We have discovered new upper constraints on path energy in this study, expressed in terms of a graph's maximum degree. Additionally, a relationship between a graph's energy and path energy is given.

Figures (1)

• Figure 1: M

Theorems & Definitions (24)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Corollary 1.5
• Corollary 1.6
• proof
• Proposition 1.7
• proof