Positive and negative 3-energies of graphs

Abstract: For a simple graph $G$ with $n$ vertices, let $A_G$ denote the adjacency matrix of $G$, and let $λ1(G) \geq λ_2(G) \geq \dots \geq λ_n(G)$ be its eigenvalues. For an integer $p \geq 2$, the positive $p$-energy and negative $p$-energy of $G$, denoted $\mathcal{E}^+_p(G)$ and $\mathcal{E}^-_p(G)$, are defined as follows: $\mathcal{E}^+_p(G) = \sum{λi(G) > 0} |λ_i(G)|^p$ and $\mathcal{E}^-_p(G) = \sum{λ_i(G) < 0} |λ_i(G)|^p,$ respectively. Tang, Liu, and Wang proposed a conjecture that, for any integer $p \geq 2$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^+_p(G) \geq \mathcal{E}^+_p(P_n)$. Akbari, Kumar, Mohar, and Pragada conjectured that, for any $p \geq 2$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^-_p(G) \geq \mathcal{E}^-_p(K_n)$, and they proved this conjecture for $p \geq 4$. In this paper, we prove that every connected $n$-vertex graph, except for $K_1$, $K_2$, and $P_3$, satisfies $\mathcal{E}^+_3(G) \geq \frac{\sqrt{5}}{2}n$. Moreover, we show that for any integer $p \geq 3$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^-_p(G) \geq \mathcal{E}^-_p(K_n)$, which improves upon the previously known result.

• The paper proves a tight lower bound for negative 3-energy in connected graphs, with equality if and only if the graph is complete.
• The methodology employs block decomposition, spectral interlacing, and explicit computations of small induced subgraphs to establish energy bounds.
• The work extends extremal energy results to p ≥ 3, bridging classical graph energy with nonlinear Schatten-p norm measurements.

Positive and Negative 3-Energies of Graphs: Tight Extremal Bounds

Introduction and Context

This paper investigates tight bounds for higher-order graph energies, specifically focusing on the positive and negative $3$-energies of simple, connected graphs. Let $A_G$ denote the adjacency matrix of a simple graph $G$ with eigenvalues $\lambda_1 \geq \dots \geq \lambda_n$. For integer $p \geq 2$, the positive $p$-energy and negative $p$-energy are defined as

$$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$

extending the classic notion of graph energy to nonlinear (Schatten-$p$) spectral sums. Not only do these parameters capture fine-grained spectral properties, but they also connect to coloring parameters and extremal combinatorics, as seen in recent work [ELPHICK2026104252].

Previous literature established several conjectures and partial results:

• For $p=2$, it was conjectured and partially verified that for any connected $A_G$0-vertex graph $A_G$1, $A_G$2 [elphick2016conjectured].
• For general $A_G$3, two conjectures posited that $A_G$4 is minimized by the path $A_G$5 and $A_G$6 is minimized by the complete graph $A_G$7 among all connected $A_G$8-vertex graphs.
• The negative $A_G$9-energy conjecture had previously been proved only for $G$0 [AKBARI202596].

This paper decisively advances our understanding for $G$1, establishing the conjectured lower bound for negative 3-energy for all $G$2-vertex connected graphs, and proving an explicit lower bound for the positive 3-energy for most graph families, accompanied by precise structural characterizations and inductive reductions.

Main Contributions

1. Negative 3-Energy Lower Bound

The authors prove that for all connected graphs $G$3 on $G$4 vertices, the negative $G$5-energy satisfies

$G$6

with equality if and only if $G$7. They further strengthen the result for graphs not isomorphic to $G$8 or $G$9, showing that $\lambda_1 \geq \dots \geq \lambda_n$0.

The proof builds on block-matrix additivity of $\lambda_1 \geq \dots \geq \lambda_n$1-energy [AKBARI202596], a refined combinatorial understanding of how induced subgraph energies interact, and a careful inductive structure analysis. The paper elaborates a host of structural lemmas:

• Any counterexample must have $\lambda_1 \geq \dots \geq \lambda_n$2, be highly constrained in connectivity, and admit only certain configurations of small induced paths and cliques.
• The minimal counterexamples are shown to be reducible to unions involving $\lambda_1 \geq \dots \geq \lambda_n$3s and cliques, which are then excluded via explicit calculations verified computationally up to $\lambda_1 \geq \dots \geq \lambda_n$4.
• The argument crucially exploits spectral interlacing and polynomial root bounds to handle various small graphs and possible non-clique substructures.

2. Positive 3-Energy Lower Bound

For all connected $\lambda_1 \geq \dots \geq \lambda_n$5-vertex graphs except $\lambda_1 \geq \dots \geq \lambda_n$6 and $\lambda_1 \geq \dots \geq \lambda_n$7, the positive 3-energy obeys

$\lambda_1 \geq \dots \geq \lambda_n$8

with $\lambda_1 \geq \dots \geq \lambda_n$9 attaining equality; thus, the bound is tight up to the exclusion of some small graphs.

The proof here leverages block-decomposition inequalities, careful case analysis based on components and edge-cuts, and lower bounds derived from induction and convexity. Computational enumeration up to $p \geq 2$0 is used to settle the base cases. The existence of "bad" small graphs like $p \geq 2$1 is shown to be the only significant obstruction; once excluded, the inductive lower bound is globally valid.

3. Generalization to Larger $p \geq 2$2

The authors show by interpolation (via Hölder-type inequalities) that the lower bound for $p \geq 2$3 extends to all $p \geq 2$4, reducing the known previous threshold of $p \geq 2$5. Thus, for any $p \geq 2$6,

$p \geq 2$7

with the $p \geq 2$8 case remaining markedly more challenging.

4. Comprehensive Structural Analysis

The paper's technical backbone consists of a careful classification of possible local configurations violating the lower bounds, quantifying energies of all small induced subgraphs via explicit spectral computations, and showing all exceptional cases are reducible. Substantial effort is devoted to handling the path $p \geq 2$9, which has energy between $p$0 and $p$1 but is unique in complicating the inductive step for $p$2, in contrast to $p$3.

Numerical and Spectral Highlights

• For $p$4, $p$5.
• For $p$6, $p$7.
• For $p$8, $p$9.
• The inequality $p$0 is exact for $p$1.

The algorithmic enumeration and computations of spectra for all graphs up to $p$2 ensure that the arguments are exhaustive for the small graph regime. All results hold universally for $p$3 by induction.

Implications and Future Directions

This work resolves, for $p$4, a pair of natural extremal graph energy conjectures that have attracted consistent attention, thereby tightening the landscape of possible extremal graphs in spectral graph theory. The result offers direct implications for questions relating to the Schatten-$p$5 norm extremals among graphs and aligns with distinct behavior observed for $p$6 versus $p$7.

From the perspective of future work:

• The $p$8 case for negative energy remains resistant to complete classification, as all paths $p$9 become difficult base cases.
• The positive energy conjecture (that, for all $$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$0, $$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$1) remains unresolved for full generality. The present methods might serve as a template for subsequent advances.
• Potential extensions could target fractional or real-valued $$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$2 (i.e., the region $$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$3), for which the present framework could provide foundational bounds.
• Applications to coloring numbers, fractional chromatic number, and norm-based graph invariants suggest broad combinatorial utility [ELPHICK2026104252, NIKIFOROV201682].

Conclusion

The paper delivers a comprehensive resolution of the conjectured extremal lower bound for negative $$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$4-energy for $$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$5 in connected graphs, together with a nearly tight bound for positive $$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$6-energy. The results are substantiated by a blend of spectral inequalities, block partitioning, explicit spectral calculations, and inductive combinatorial arguments. The careful treatment of small graph obstructions, especially $$\mathcal{E}^+_p(G) = \sum_{\lambda_i > 0} |\lambda_i|^{p}, \qquad \mathcal{E}^-_p(G) = \sum_{\lambda_i < 0} |\lambda_i|^{p},$$7, and the general methodology deployed, set a new standard for future analyses in the area of nonlinear spectral graph parameters.

Reference: "Positive and negative 3-energies of graphs" (2604.15656)