Some Bond Incident Degree Indices of (Molecular) Graphs with Fixed Order and Number of Cut Vertices

Junwei Wang,Tayyba Zafar,Abeer M. Albalahi,Wael W. Mohammed,Akbar Ali

doi:10.1155/2021/9970254

Abstract

A bond incident degree (BID) index of a graph G is defined as ∑ u v ∈ E G f d G u , d G v , where d G w denotes the degree of a vertex w of G , E G is the edge set of G , and f is a real-valued symmetric function. The choice f d G u , d G v = a d G u + a d G v in the aforementioned formula gives the variable sum exdeg index SEI a , where a ≠ 1 is any positive real number. A cut vertex of a graph G is a vertex whose removal results in a graph with more components than G has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by V n , k the class of all n -vertex graphs with k ≥ 1 cut vertices and containing at least one cycle. Recently, Du and Sun [AIMS Mathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of SEI a from the set V n k for a > 1 . In the present paper, we not only characterize the graphs with the minimum value of SEI a from the set V n k for a > 1 , but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all n -vertex molecular graphs with k ≥ 1 cut vertices and containing at least one cycle.

Highlights

IntroductionGraph invariants of the following form are known as the bond incident degree (BID) indices [1–4]: BID(G) 􏽘 f dG(u), dG(v)􏼁,
Graph invariants of the following form are known as the bond incident degree (BID) indices [1–4]: BID(G) 􏽘 f dG(u), dG(v)􏼁, [1] uv∈E(G)where dG(w) denotes the degree of a vertex w of the graph G, E(G) is the edge set of G, and f is a real-valued symmetric function
Lemma 4 guaranties that Gmin does not contain a pendent vertex having a neighbor w of degree greater than 2 such that w remains a cut vertex in Gmin − v

Summary

Introduction

Graph invariants of the following form are known as the bond incident degree (BID) indices [1–4]: BID(G) 􏽘 f dG(u), dG(v)􏼁,. If a ≠ 1 is a positive real number, the variable sum exdeg index SEIa of a graph G can be defined as SEIa(G) 􏽘 dG(v)adG(v). Let k n be of all n-vertex graphs with k ≥ 1 cut vertices and containing at least one cycle. Since the cycle graph Cn of order n has no cut vertex and it is the only unicyclic graph of minimum degree at least 2, the result is an immediate consequence of Lemma 3. Vr in a graph G is called a pendent path if one of the two vertices v1, vr, is pendent and the other has degree greater than 2, and every other vertex (if exists) of P has degree 2. Vr is a pendent path in which v1 has degree greater than 2, v1 is known as the branching vertex. Let xy ∈ E(G) be an edge lying on the unique cycle of

It holds that

Main Result