The total irregularity of a simple undirected graph G is defined as 1/2×Σu,v∈V(G)|dG(u)−dG(v|, where dG(u) denotes the degree of a vertex u∈V(G). General graphs with maximal total irregularity were characterized by Abdo, Brandt, and Dimitrov in 2014. Here, we extend this result by characterizing graphs with maximal total irregularity among the graphs with a given number n of vertices and m of edges, both in the general and in the connected case. Since a connected graph has cyclomatic number c = m−n+1, the results presented here also generalize those of You, Yang, and You from 2014, who characterized graphs with cyclomatic numbers 1 and 2 having maximal total irregularity. In addition, we reformulate the problem as a combinatorial optimization task and provide its solution in this context.
Last update: November 19, 2025.