Some Vertex/Edge-Degree-Based Topological Indices of  r -Apex Trees

Akbar Ali,Qasim Ali Chaudhry,Ekram E Ali,Jia-Bao Liu,Zahid Raza,Farooq Ahmad,Waqas Iqbal,Muhammad Kamran Siddiqui

doi:10.1155/2021/4349074

Abstract

In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph G , its vertex-degree-based topological indices of the form BID G = ∑ u v ∈ E G β d u , d v are known as bond incident degree indices, where E G is the edge set of G , d w denotes degree of an arbitrary vertex w of G , and β is a real-valued-symmetric function. Those BID indices for which β can be rewritten as a function of d u + d v − 2 (that is degree of the edge u v ) are known as edge-degree-based BID indices. A connected graph G is said to be r -apex tree if r is the smallest nonnegative integer for which there is a subset R of V G such that R = r and G − R is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary BID index from the class of all r -apex trees of order n , where r and n are fixed integers satisfying the inequalities n − r ≥ 2 and r ≥ 1 .

Highlights

All the graphs discussed in the present paper are finite. e vertex set and edge set of a graph G are denoted by V(G) and E(G), respectively
Every tree is a 0apex tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary bond incident degree (BID) index from the class of all r-apex trees of order n, where r and n are fixed integers satisfying the inequalities n − r ≥ 2 and r ≥ 1
(i) If for every connected noncomplete graph G, the inequality I(G + e) > I(G) holds for every e ∉ E(G); the graph attaining the maximum value of the topological index I among all r-apex trees of order n is isomorphic to the join Kr + T, where r and n are fixed integers satisfying the inequalities r ≥ 1 and n − r ≥ 2 and T is a tree of order n − r

Summary

Introduction

All the graphs discussed in the present paper are finite. e vertex set and edge set of a graph G are denoted by V(G) and E(G), respectively. If BID(G) 􏽐uv∈E(G)β(du, dv) is a bond incident degree index such that, for every connected noncomplete graph H, the inequality BID(H + e) > BID(H) holds for every e ∉ E(H); Kr + Sn−r uniquely attains the maximum BID index among all r-apex trees of order n, where r and n are fixed integers satisfying the inequalities r ≥ 1 and n − r ≥ 2.

Results

Conclusion