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The concept of the Sombor indices of a graph was introduced by Gutman. A vertex-edge variant of the Sombor index of graphs is called the KG-Sombor index. Recently, the Sombor and KG-Sombor indices of Kragujevac trees were studied, and the extremal Kragujevac trees with respect to these indices were empirically determined. Here we give analytical proof of the results. [ABSTRACT FROM AUTHOR]

The graph invariant RM2, known under the name reduced second Zagreb index, is de_ned as RM2(G) = Σuv∈E(G) (dG(v) -- 1)(dG(v) -- 1), where dG(v) is the degree of the vertex v of the graph G. In this paper, we give a tight upper bound of RM2 for the class of graphs of order n and size m with at least one dominating vertex. Also, we obtain sharp upper bounds on RM2 for all graphs of order n with k dominating vertices and for all graphs of order n with k pendant vertices. Finally, we give a sharp upper bound on RM2 for all k-apex trees of order n. Moreover, the corresponding extremal graphs are characterized. [ABSTRACT FROM AUTHOR]