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

Quanyu Tang, Yinchen Liu, Wei Wang

TL;DR

This work extends the theory of graph energies by introducing positive and negative $p$-energies, $\mathcal{E}_p^+(G)$ and $\mathcal{E}_p^-(G)$, as sums of the $p$-th powers of the positive and negative adjacency-eigenvalues. It provides tighter lower bounds for these quantities under edge addition via $H=G-e$, exploiting majorization and Lidski inequalities, and recovers sharper $p=2$ results on the square energies $s^+(G)$ and $s^-(G)$. It then constructs explicit counterexamples showing that monotonicity of $\mathcal{E}_p^+(G)$ under edge addition fails for $1\le p<3$ using the graphs $\overline{S_{n,n}}$ and $(\overline{S_{n,n}})^+$, with asymptotic analysis establishing the non-monotonic behavior. The paper concludes with open questions and conjectures about extremal $p$-energies and potential monotonicity for larger $p$, highlighting directions for future research in spectral graph theory and connections to related bounds and invariants.

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.

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

TL;DR

This work extends the theory of graph energies by introducing positive and negative -energies, and , as sums of the -th powers of the positive and negative adjacency-eigenvalues. It provides tighter lower bounds for these quantities under edge addition via , exploiting majorization and Lidski inequalities, and recovers sharper results on the square energies and . It then constructs explicit counterexamples showing that monotonicity of under edge addition fails for using the graphs and , with asymptotic analysis establishing the non-monotonic behavior. The paper concludes with open questions and conjectures about extremal -energies and potential monotonicity for larger , highlighting directions for future research in spectral graph theory and connections to related bounds and invariants.

Abstract

In this paper, we introduce the concepts of positive and negative -energies of graphs and investigate their behavior under edge addition. Specifically, we generalize the classical notions of positive and negative square energies to the -energy setting, denoted by and , respectively. We establish improved lower bounds for these quantities under edge addition, which sharpen existing results by Abiad et al.\ in the case . Furthermore, we address the monotonicity problem for under edge addition, and construct a family of counterexamples showing that monotonicity fails for . Finally, we conclude with several open problems for further investigation.

Paper Structure

This paper contains 4 sections, 14 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Improved Bounds under Edge Addition
  3. Monotonicity for Edge Addition
  4. Concluding Remarks and Open Problems

Key Result

Theorem 1.3

Let $G$ be a graph, let $H = G - e$ (where $e$ is an edge of $G$), and let $\lambda_1 \geq \cdots \geq \lambda_n$ and $\theta_1 \geq \cdots \geq \theta_n$ denote the eigenvalues of $A(G)$ and $A(H)$, respectively. If $H$ has at least two positive eigenvalues and at least two negative eigenvalues, th

Figures (1)

  • Figure 1: The complement of double star graph $\overline{S_{n_1,n_2}}$.

Theorems & Definitions (31)

  • Conjecture 1.1: Elphick et al. EFGW16
  • Conjecture 1.2: Cioaba
  • Theorem 1.3: Abiad2023, Theorem 2.5
  • Definition 1.4
  • Remark 1.6
  • Remark 1.7
  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Corollary 2.4
  • ...and 21 more