On the Positive and Negative $p$-Energies of Graphs under Edge Addition

TL;DR

This paper introduces positive and negative $p$-energies of graphs, explores their behavior under edge addition, improves bounds, and shows monotonicity fails for certain $p$ values, opening new research directions.

Contribution

It generalizes classical graph energies to the $p$-energy setting, improves bounds for edge addition, and demonstrates non-monotonicity for specific $p$ ranges.

Findings

Established improved lower bounds for $p$-energies under edge addition.

Constructed counterexamples showing non-monotonicity for $1 \,\leq\, p < 3$.

Extended classical energy concepts to a broader $p$-energy framework.

Abstract

In this paper, we introduce the concepts of positive and negative $p$-energies of graphs and investigate their behavior under edge addition. Specifically, we generalize the classical notions of positive and negative square energies to the $p$-energy setting, denoted by $\mathcal{E}_p^{+}(G)$ and $\mathcal{E}_p^{-}(G)$, respectively. We establish improved lower bounds for these quantities under edge addition, which sharpen existing results by Abiad et al.\ in the case $p=2$. Furthermore, we address the monotonicity problem for $\mathcal{E}_p^{+}(G)$ under edge addition, and construct a family of counterexamples showing that monotonicity fails for $1 \leq p < 3$. Finally, we conclude with several open problems for further investigation.