Unified extremal results for [formula omitted]-apex unicyclic graphs (trees)

Abstract

A k-cone c-cyclic graph is the join of the complete graph Kk and a c-cyclic graph (if k=0, we get the usual connected graph). A k-apex tree (resp., k-apex unicyclic graph) is defined as a connected graph G with a k-subset Vk⊆V(G) such that G−Vk is a tree (resp., unicylic graph), but G−X is not a tree (resp., unicylic graph) for any X⊆V(G) with |X|<k. In this paper, we extend those extremal results and majorization theorems concerning connected graphs of Liu et al. (2019) to k-cone c-cyclic graphs. We also use a unified method to characterize the extremal maximum and minimum results of many topological indices in the class of k-apex trees and k-apex unicyclic graphs, respectively. The later results extend the main results of Javaid et al. (2019); Liu et al. (2020) and partially answer the open problem of Javaid et al. (2019). Except for the new majorization theorem, some new techniques are also established to deal with the minimum extremal results of this paper.