On the maximum $\sigma$-irregularity of trees with given order and maximum degree

The $\sigma$-irregularity index of a graph is defined as the sum of squared degree differences over all edges and provides a sensitive measure of structural heterogeneity. In this paper, we study the problem of maximizing $\sigma(T)$ among all trees of fixed order $n$ and prescribed maximum degree $\Delta\ge4$. By expressing the problem in terms of edge--degree multiplicities, we derive a linear programming formulation and analyze its dual. This approach yields sharp upper bounds for $\sigma(T)$ and leads to a detailed description of extremal degree--pair distributions. We show that the extremal problem can be completely resolved for the congruence classes $n\equiv1\pmod{\Delta}$ and $n\equiv0\pmod{\Delta}$. When $n\equiv1\pmod{\Delta}$, the linear program admits an integral optimal solution, and the bound for $\sigma(T)$ is tight. When $n\equiv0\pmod{\Delta}$, the linear relaxation is not attainable by any tree; nevertheless, by introducing a penalty function derived from dual slack variables, we determine the exact maximum value of $\sigma(T)$. In both cases, all extremal trees are characterized explicitly and consist exclusively of vertices of degrees $1$, $2$, and $\Delta$, with edges incident to $\Delta$-vertices playing a dominant role.