On the $\alpha$-index of minimally 2-connected graphs with given order or size

TL;DR

This paper characterizes the extremal minimally 2-connected graphs with the maximum $eta$-index for $eta o 1$ within a specific spectral matrix family, based on order or size.

Contribution

It provides a characterization of extremal graphs with maximum $eta$-index among minimally 2-connected graphs for $eta$ in [1/2,1), considering order and size.

Findings

Identifies extremal graphs with maximum $eta$-index for $eta o 1$

Characterizes these graphs based on order and size

Focuses on minimally 2-connected graphs

Abstract

For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. The largest eigenvalue of $A_\alpha(G)$ is called the $\alpha$-index or the $A_\alpha$-spectral radius of $G$. A graph is minimally $k$-connected if it is $k$-connected and deleting any arbitrary chosen edge always leaves a graph which is not $k$-connected. In this paper, we characterize the extremal graphs with the maximum $\alpha$-index for $\alpha \in [\frac{1}{2},1)$ among all minimally 2-connected graphs with given order or size, respectively.