Conjectured bound involving the maximum degree of a tree

Establish, for every tree T with n vertices and m edges and every integer k satisfying 0<k<5, the inequality 0≤2^k(n+m)^2/[2(k+1)Δ(Δ−1)^2]≤2k, where Δ is the maximum degree of T, thereby validating the conjectured estimate used to derive a lower bound for the Sigma index.

Background

Section 4 develops lower bounds for the Sigma index of trees in terms of degree sequences and structural parameters. The authors introduce a conjecture concerning the order n, size m, and maximum degree Δ of a tree, with an integer parameter k restricted to 0<k<5.

The conjectured inequality is then used in the proof of Proposition~\ref{manProtun2} to control the quantity (n+m)2/[Δ(Δ−1)2] and derive a further lower bound for the Sigma index. Although the manuscript includes an attempted justification immediately after the conjecture, the result is explicitly presented as a conjecture and is therefore the sole unresolved statement identified under the requested inclusion criteria.

References

Based on Conjecture~\ref{conjecturen1} which establishes Proposition~\ref{manProtun2}, this conjecture is employed to refine the optimal terms that play a crucial role in enhancing the optimal behavior of the Sigma index.

\begin{conjecture}~\label{conjecturen1} For any tree $T$ with $n$ vertices and $m$ edges. Let $k$ be an integer where $0<k<5$. Then, \begin{equation}~\label{eq1conjecturen1} 0\leqslant \dfrac{2k(n+m)2}{2(k+1)\Delta(\Delta-1)2} \leqslant 2k. \end{equation} \end{conjecture}

— Behavior of The Extremal Bounds on the $σ$-Irregularity  (2510.06845 - Hamoud et al., 8 Oct 2025) in Conjecture 1 (Conjecture~\ref{conjecturen1}), Section 4, immediately before Proposition~\ref{manProtun2}