Sombor indices–back to geometry

Abstract

The Sombor index (\(SO\)) is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices of \(\sqrt{d_i^2+d_j^2}\), where \(d_i\) is the degree of the \(i\)-th vertex. It has been conceived using geometric considerations. Numerous researches of \(SO\) that followed, ignored its geometric origin. We now show that geometry-based reasonings reveal the geometric background of several classical topological indices (Zagreb, Albertson) and lead to a series of new \(SO\)-like degree-based graph invariants.