Maximum and minimum Sombor index among k-apex unicyclic graphs and k-apex trees

Abstract

The Sombor index [Formula: see text] of a graph [Formula: see text] is defined as [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text] of [Formula: see text]. A [Formula: see text]-cone [Formula: see text]-cyclic graph is the join of the complete graph [Formula: see text] and a connected [Formula: see text]-cyclic graph. A [Formula: see text]-apex tree (respectively, [Formula: see text]-apex unicyclic graph) is defined as a connected graph [Formula: see text] with a [Formula: see text]-subset [Formula: see text] such that [Formula: see text] is a tree (respectively, unicyclic graph), but [Formula: see text] is not a tree (respectively, unicyclic graph) for any [Formula: see text] with [Formula: see text]. In this paper, we show the minimal graphs of [Formula: see text] among all [Formula: see text]-cone [Formula: see text]-cyclic graphs with [Formula: see text] as their degree sequence, and determine the extremal values and extremal graphs of [Formula: see text] among [Formula: see text]-apex unicyclic graphs and [Formula: see text]-apex trees, respectively.