On the σt-irregularity and the inverse irregularity problem

Abstract

The σ-irregularity index is a natural variant of the well-established Albertson irregularity index. Here, we introduce an irregularity measure based on the σ-irregularity, which is a graph invariant with respect to a given degree sequence. We define it as σt(G)=12∑v,w∈V(G)(dG(v)−dG(w))2, where dG(v) is the degree of a vertex v of G, and named it the total σ-irregularity. We characterize irregular graphs with minimal σt-irregularity. In addition, we consider the so-called inverse problem for the Albertson irregularity index, the total irregularity, and the σt-irregularity. For those irregularity measures, we study the problem for general graphs, trees, and c-cyclic graphs.