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arxiv: 2605.24668 · v1 · pith:32A5STYGnew · submitted 2026-05-23 · 🧮 math.CO

A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs

Bo Ning , Jing Zeng This is my paper

Pith reviewed 2026-06-30 12:59 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C50
keywords unicyclic graphadjacency matrixeigenvaluessquare sumpositive eigenvaluesnegative eigenvaluesconjecture
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The pith

Unicyclic graphs with odd cycle length k satisfy s+(G) > n > s-(G) if k ≡ 3 mod 4 and s+(G) < n < s-(G) if k ≡ 1 mod 4.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes the truth of a conjecture on the sums of the squares of positive and negative adjacency eigenvalues in unicyclic graphs. The conjecture states that for a graph with a single odd-length cycle of length k, the relative sizes of these sums compared to the number of vertices n are fixed by k modulo 4. The proof confirms both directions of the inequality depending on whether k is congruent to 1 or 3 modulo 4. Readers interested in spectral graph theory would see this as resolving how the unique cycle influences the eigenvalue distribution in these graphs.

Core claim

Let G be a unicyclic graph of order n with unique cycle of odd length k. Then if k ≡ 3 (mod 4), s+(G) > n > s-(G); if k ≡ 1 (mod 4), s+(G) < n < s-(G). The authors prove this statement holds for every such graph G.

What carries the argument

The decomposition of the unicyclic graph into its cycle and attached trees, allowing the eigenvalue sums to be analyzed by separating the contributions from the cycle eigenvalues and the tree adjustments.

Load-bearing premise

The square sums s+ and s- for the full graph can be written as the corresponding sums for the cycle plus non-negative or controlled terms from the trees that preserve the sign of the difference from n.

What would settle it

A counterexample would be any unicyclic graph with k=5 where s+(G) is not less than n, or explicit calculation of all eigenvalues for the cycle C_5 with an attached path showing the sums violate the inequality.

read the original abstract

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues, respectively. Akbari, Kumar, Mohar, Pragada, and Zhang conjectured that, when $k$ is odd, the value of $k$ modulo $4$ determines which of $s^+(G)$ and $s^-(G)$ is greater than $n$. More precisely, if $k\equiv 3\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\equiv 1\pmod 4$, then $s^+(G)<n<s^-(G)$. We confirm this conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves the conjecture of Akbari et al. that for any unicyclic graph G on n vertices whose unique cycle has odd length k, the sums s+(G) and s-(G) of the squares of the positive and negative adjacency eigenvalues satisfy: if k ≡ 3 (mod 4) then s+(G) > n > s-(G), while if k ≡ 1 (mod 4) then s+(G) < n < s-(G). The argument proceeds by decomposing G into its cycle plus attached trees and reducing the eigenvalue sums to quantities determined by the cycle length modulo 4.

Significance. The result settles a concrete conjecture in spectral graph theory for the entire class of unicyclic graphs. It supplies an explicit, case-by-case comparison of s+ and s- to the order n that depends only on the parity and residue class of the cycle length, thereby giving a complete classification for this family.

minor comments (2)
  1. The notation for the positive and negative square sums is introduced in the abstract but should be restated explicitly in the first paragraph of §2 before any calculations begin.
  2. Lemma 3.2 states that the contribution of each pendant path is independent of the attachment vertex; a one-sentence reminder of the underlying matrix-block argument would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript and for recommending acceptance. The referee's summary correctly describes the main result: a proof of the conjecture of Akbari et al. for all unicyclic graphs with an odd cycle.

Circularity Check

0 steps flagged

No circularity: proof of externally stated conjecture

full rationale

The manuscript proves a conjecture originally posed by Akbari, Kumar, Mohar, Pragada, and Zhang (distinct authors) on the sign-dependent comparison of s+(G) and s-(G) versus n for unicyclic graphs with odd cycle length k. The argument proceeds from the standard structural decomposition of any unicyclic graph into a cycle plus pendant trees, a fact independent of the target conjecture and not derived within the paper. No equations reduce a claimed prediction to a fitted input by construction, no uniqueness theorem is imported via self-citation, and no ansatz is smuggled through prior work by the same authors. The derivation therefore remains self-contained against the external conjecture statement and does not collapse to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a proof in spectral graph theory and therefore rests on standard linear-algebra facts about real symmetric matrices and combinatorial reductions for unicyclic graphs; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Adjacency matrix of a graph is real symmetric and therefore has real eigenvalues whose squares sum to twice the number of edges.
    Invoked implicitly when defining s+ and s- and comparing them to n.

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Reference graph

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