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.

翻译：σ-不规则性指标 $σ(G) = \sum_{uv \in E(G)} (d_G(u) - d_G(v))^2$ 用于度量图中沿各边的总度数不平衡性。我们在固定阶数 $n$ 且最大度 $Δ= 6$ 的树类中研究 $σ(T)$ 的极值问题。通过结合握手恒等式与同余论证的惩罚函数框架，我们确定了 $n$ 模 $6$ 的每个剩余类中 $σ(T)$ 的精确最大值，并证明惩罚函数可能的最小值为 $0, 10, 20, 22, 30,$ 和 $40$。针对每种情况，我们根据度数计数与边重数完整刻画了所有最大化树的结构。在六个剩余类中的五类里，所有极值树仅包含度数为 $1, 2,$ 和 $6$ 的顶点；而对于 $n \equiv 3 \pmod{6}$ 的情形，则出现了一类额外涉及度数为 $3$ 的顶点的例外族。这些结果扩展了先前关于较小度约束下σ-不规则性的研究，并说明了随着最大度的增加，该问题的组合复杂性迅速增长。