Non-semiregular bipartite graphs with integer Sombor index

Abstract

Abstract For a simple graphG, the Sombor index (a recently introduced vertex-degree based molecular structure descriptor) is defined as SO(G) = ∑ uv∈E(G) √ du + dv, where dv is the degree of v. A graph is bipartite semi-regular if it is bipartite bidegreed and all vertices in the same class of bipartition have the same degree. In the recent paper [T. Došlić, T. Réti, A. Ali, Discrete Math. Lett. 7 (2021) 1–4] the following claim was posed: “Let G be a connected bipartite graph. Then SO(G) is an integer if and only if G is bipartite semi-regular and its degrees δ and ∆ appear as non-maximal elements in some Pythagorean triple”. In the present paper we show that the ‘only if’ part of the mentioned claim is not true. More precisely, we construct infinite number of connected bipartite graphs such that in their degree sequences there are three or four distinct numbers.