Trees with maximum $σ$-irregularity under a prescribed maximum degree 6

TL;DR

This work determines the maximal σ-regularity index $σ(T)$ among all $n$-vertex trees with maximum degree $Δ=6$ by reducing the problem to minimizing a penalty function $P(T)$ over edge-type counts via handshake identities and congruence analysis. The authors establish six residue-class regimes for $n\pmod{6}$, with corresponding minimal penalties $P_{ ext{min}}(n)\in\{0,10,20,22,30,40\}$, and provide complete characterizations of all extremal trees, usually using degrees $\{1,2,6\}$, except for an exceptional pattern when $n\equiv 3\pmod{6}$ that involves degree-$3$ vertices. They supply explicit algebraic descriptions of minimizers (edge-count vectors $m_{i,j}$ and degree counts $n_i$) and constructive recursive families that generate all extremal trees for each congruence class. The results extend prior Δ-bound studies and illustrate the increasing combinatorial complexity of σ-extremal trees as the maximum degree grows, suggesting directions for extending the framework to larger $Δ$.

Abstract

The sigma-irregularity index $σ(G) = \sum_{uv \in E(G)} (d_G(u) - d_G(v))^2$ measures the total degree imbalance along the edges of a graph. We study extremal problems for $σ(T)$ within the class of trees of fixed order $n$ and bounded maximum degree $Δ= 6$. Using a penalty-function framework combined with handshake identities and congruence arguments, we determine the exact maximum value of $σ(T)$ for every residue class of $n$ modulo $6$, showing that the possible minimum values of the penalty function are $0, 10, 20, 22, 30,$ and $40$. For each case, we provide a complete characterization of all maximizing trees in terms of degree counts and edge multiplicities. In five of the six residue classes, all extremal trees contain only vertices of degrees $1, 2,$ and $6$, while for $n \equiv 3 \pmod{6}$ an additional exceptional family arises involving vertices of degree $3$. These results extend earlier work on sigma-irregularity for smaller degree bounds and illustrate the rapidly growing combinatorial complexity of the problem as the maximum degree increases.

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