On the Extremal Trees for Some Bond Incident Degree Indices with a Fixed Number of Segments

Muzamil Hanif,Akhlaq Ahmad Bhatti,Ebenezer Bonyah,Muhammad Javaid,Hani Shaker

doi:10.1155/2022/4032709

Abstract

The Gourava indices and hyper-Gourava indices are introduced by Kulli in 2017. These graph invariants are related to the degree of vertices of a graph G . Let T n , r be the class of all n − vertex chemical trees with r segments. In this paper, we characterize the graphs with the maximum value of the above indices in the class of chemical trees. In addition to different degree sequences, sharp upper bounds on those indices of trees with a fixed number of segments are determined, and the corresponding extremal trees are characterized.

Highlights

M1(G) 􏽘 d2v(G), v∈V(G) (1)M2(G) 􏽘 du(G)dv(G).u∼v e first Zagreb index is defined as [4]M1(G) 􏽘 du(G) + dv(G)􏼁. (2)u∼v e extremal Zagreb indices have been studied for certain classes of trees in last some decades
U∼v e extremal Zagreb indices have been studied for certain classes of trees in last some decades
Goubko and Gutman [16] characterized the trees with the minimum first Zagreb index among the trees with a fixed Journal of Chemistry number of pendent vertices

Summary

Introduction

Lin [17] characterized the tree that can maximize and minimize the first Zagreb index with fixed number of segments. Borovicanin, Das, Furtula, and Gutman [18, 19] characterized the tree with maximum and minimum Zagreb indices for certain classes of trees with fixed number of segments or branching vertices. We characterize the maximum Gourava indices and hyper-Gourava indices among all chemical trees of prescribed degree sequences with fixed number of segments. Let Tn,r be the set of all classes of chemical trees with order n and exactly r segments. Recall 1Tn,r be a class of all n− vertex chemical trees of degree sequence (r, 􏽼2√,√2√√, 􏽻.􏽺.√.√,√√2􏽽, 􏽼1√,√1√√, .􏽻􏽺.√.√,√√1􏽽). Ere exists a vertex w ∈ 1Tmax of degree two, such that w ∼ w1 and w ∼ w2, where w1, w2 are non-pendent vertices, i.e., On the contrary, that u ∼ v such that u and v are pendent and branching vertices, respectively. ere exists a vertex w ∈ 1Tmax of degree two, such that w ∼ w1 and w ∼ w2, where w1, w2 are non-pendent vertices, i.e.,

We have

1Tmax max also x

Conclusions