Trees with maximum $\sigma$-irregularity under a prescribed maximum degree 6

TL;DR

This paper determines the maximum sigma-irregularity index for trees with fixed order and maximum degree 6, characterizing extremal trees based on degree distributions and residue classes.

Contribution

It provides exact maximum values and characterizations of extremal trees for each residue class of n modulo 6, extending previous work to higher degree bounds.

Findings

Maximum sigma-irregularity values are identified for each residue class.

Extremal trees mostly contain vertices of degrees 1, 2, and 6.

An exceptional family appears for n ≡ 3 mod 6 involving degree 3 vertices.

Abstract

The sigma-irregularity index $\sigma(G) = \sum_{uv \in E(G)} (d_G(u) - d_G(v))^2$ measures the total degree imbalance along the edges of a graph. We study extremal problems for $\sigma(T)$ within the class of trees of fixed order $n$ and bounded maximum degree $\Delta = 6$. Using a penalty-function framework combined with handshake identities and congruence arguments, we determine the exact maximum value of $\sigma(T)$ 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.