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On the Maximum Atom-Bond Sum-Connectivity Index of Graphs

Tariq Alraqad, Hisham Saber, Akbar Ali, Abeer M. Albalahi

Abstract

The atom-bond sum-connectivity (ABS) index of a graph $G$ with edges $e_1,\cdots,e_m$ is the sum of the numbers $\sqrt{1-2(d_{e_i}+2)^{-1}}$ over $1\le i \le m$, where $d_{e_i}$ is the number of edges adjacent with $e_i$. In this paper, we study the maximum values of the ABS index over graphs with given parameters. More specifically, we determine the maximum ABS index of connected graphs of a given order and with a fixed (i) minimum degree, (ii) maximum degree, (iii) chromatic number, (iv) independence number, or (v) number of pendent vertices. We also characterize the graphs attaining the maximum ABS values in all of these classes.

On the Maximum Atom-Bond Sum-Connectivity Index of Graphs

Abstract

The atom-bond sum-connectivity (ABS) index of a graph with edges is the sum of the numbers over , where is the number of edges adjacent with . In this paper, we study the maximum values of the ABS index over graphs with given parameters. More specifically, we determine the maximum ABS index of connected graphs of a given order and with a fixed (i) minimum degree, (ii) maximum degree, (iii) chromatic number, (iv) independence number, or (v) number of pendent vertices. We also characterize the graphs attaining the maximum ABS values in all of these classes.
Paper Structure (2 sections, 9 theorems, 34 equations)

This paper contains 2 sections, 9 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Results

Key Result

Lemma 1

Let $G$ be a graph. If $u$ and $v$ are non-adjacent vertices in $G$ then $ABS(G + uv) > ABS(G)$.

Theorems & Definitions (9)

  • Lemma 1: see g10
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5