On extremal graphs with respect to the ABS index

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 κ. In this paper, we provide complete solutions to all of these problems.