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Extremal values for the square energies of graphs

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arxiv 2409.15504 v2 pith:D3KHGVOR submitted 2024-09-23 math.CO math.SP

Shengtong Zhang This is my paper · ORCID
classification math.COmath.SP
keywords squareconjectureedgeselphickenergiesextremalgammagraph
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abstract

Let $G$ be a graph with $n$ non-isolated vertices and $m$ edges. The positive / negative square energies of $G$, denoted $s^+(G)$ / $s^-(G)$, are defined as the sum of squares of the positive / negative eigenvalues of the adjacency matrix $A_G$ of $G$. In this work, we provide several new tools for studying square energy encompassing semi-definite optimization, graph operations, and surplus. Using our tools, we prove the following results on the extremal values of $s^{\pm}(G)$ with a given number of vertices and edges. 1. We have $\min(s^+(G), s^-(G)) \geq n - \gamma \geq \frac{n}{2}$, where $\gamma$ is the domination number of $G$. This verifies a conjecture of Elphick, Farber, Goldberg and Wocjan up to a constant, and proves a weaker version of this conjecture introduced by Elphick and Linz. 2. We have $s^+(G) \geq m^{6/7 - o(1)}$ and $s^-(G) = \Omega(m^{1/2})$, with both exponents being optimal.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A positive square-energy strengthening of Tur\'an's theorem

    math.CO 2026-07 conditional novelty 8.0 of 10

    Every n-vertex graph with clique number ω has √s⁺(G) ≤ (1−1/ω)n, where s⁺(G) is the sum of squared positive adjacency eigenvalues.

  2. The positive and negative square-energy conjecture

    math.CO 2026-07 accept novelty 8.0 of 10

    Every connected graph G on n vertices satisfies min{s+(G), s−(G)} ≥ n−1, confirming the Elphick–Farber–Goldberg–Wocjan conjecture.

  3. Refinement of a conjecture on positive square energy of graphs

    math.CO 2025-06 conditional novelty 8.0 of 10

    For connected claw-free graphs with maximum degree at least 3 and for diameter-2 graphs other than stars and C5, the positive square energy is at least the number of vertices.

  4. A graph energy conjecture through the lenses of semidefinite programming

    math.CO 2025-09 reject novelty 6.0 of 10

    New SDP-based bounds relate graph energy to the fractional clique cover number, Hoffman's ratio number, and Schrijver's theta number, supporting a 40-year-old conjecture without proving it.

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