Extremal trees for the modified first Zagreb connection index with fixed number of segments or vertices of degree 2

Abstract

The modified first Zagreb connection index $ZC_{1}^{*} $ is a graph invariant, initially appeared within a formula of the total electron energy of alternant hydrocarbons in 1972, and revisted in a recent paper [A. Ali, N. Trinajstić. A novel/old modification of the first Zagreb index. Mol Inform. 2018;37(6). Art# 1800008; arXiv:1705.10430 [math.CO]]. In this paper, the graph(s) with the maximum/minimum $ZC_{1}^{*} $ value is/are characterized from the class of all n-vertex trees with fixed number of segments. As the number of segments in a tree can be determined from the number of vertices of degree 2 (and vice versa), the trees with the extremum $ZC_{1}^{*} $ values are also determined from the class of all n-vertex trees having a fixed number of vertices of degree 2.