On extremal graphs with respect to the ABS index

TL;DR

This paper solves open problems about identifying extremal graphs that maximize the ABS index within various classes, including graphs with cut-vertices, k-partiteness, and fixed vertex connectivity.

Contribution

It provides complete characterizations of extremal graphs for the maximum ABS index in several specific graph classes, addressing previously open questions.

Findings

Characterization of graphs with maximum ABS index among those with p cut-vertices.

Identification of extremal graphs with given k-partiteness.

Solutions for extremal bipartite graphs with fixed vertex connectivity.

Abstract

Recently, Ali et al. posed several open problems concerning extremal graphs with respect to the ABS index. These problems involve characterizing graphs that attain the maximum ABS index within specific graph classes, including: connected graphs with n vertices and p cut-vertices; (ii) connected graphs of order n with vertex k-partiteness; and (iii) connected bipartite graphs of order n with a fixed vertex connectivity \kappa. In this paper, we provide complete solutions to all of these problems.