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

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 $n\equiv 3(\mathrm{mod}6)$ 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.