A complete characterization of maximally σ-irregular trees with prescribed maximum degree

Abstract

The sigma-irregularity of a graph G = (V, E) is defined as the sum, over all edges uv in E, of (d(u) - d(v))^2, where d(u) denotes the degree of vertex u. A tree on n vertices with maximum degree Delta is called maximal if it attains the greatest possible sigma-irregularity among all such trees. The maximal trees are already known for chemical trees (Delta <= 4) and for Delta = 5. In this paper, we characterize the maximal trees for every Delta >= 6 when n >= Delta(Delta - 1) + 1. We introduce three families of trees, T'{n,Delta}, T''{n,Delta}, and T'''{n,Delta}. All trees within the same family have the same sigma-irregularity, and we derive an explicit formula for the value attained by each family. Comparing these formulas determines which family has the greatest sigma-irregularity for given n and Delta. We then prove that a tree is maximal if and only if it belongs to a family attaining this greatest value. In contrast to the case Delta <= 5, the two natural candidate families T'{n,Delta} and T''{n,Delta} are not sufficient, since for every Delta >= 7 and every n congruent to 3 modulo Delta, the trees in T'''{n,Delta} have strictly greater sigma-irregularity.