On extremal graphs with respect to the ABS index

TL;DR

This work resolves three open extremal problems for the atom-bond sum connectivity index $ABS$. It first shows that extremals among connected graphs with $n$ vertices and $p$ cut-vertices must have clique-like blocks with each cut-vertex in exactly two blocks, and the maximum is achieved by the graph $\mathbb{K}_n^{p}$ (up to the path exception). For vertex $k$-partiteness, Turán graphs $T(n,k)$ maximize $ABS$ among complete $k$-partite graphs, and in the wider class the maximum occurs at $K_r \vee T(n-r,k)$, i.e., the join of a clique with a Turán graph. In bipartite graphs with fixed vertex connectivity $\kappa$, the authors identify a detailed structural family, with the maximum achieved either by $K_{\kappa,n-\kappa}$ (when $\kappa$ is near $n/2$) or by the bipartite family $\overline{\mathcal{K}}_{\kappa}[x,y]$ in explicit parity- and parameter-dependent forms. Collectively, these results deliver complete solutions to several open extremal questions for the $ABS$ index and provide precise graph-structural characterizations of the maximizers.

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.

Figures (3)

• Figure 2.1: Graphs $\mathbb{K}_{n}^{p}$, for $n\ge 2p$ (left) and $n<2p$ (right).
• Figure 4.1: The graphs discussed in Remark \ref{['fremark']}.
• Figure 4.2: Graph $\overline{\mathcal{K}}_{\kappa}[x,y]$

Theorems & Definitions (28)

• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• Theorem 3.3
• proof