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arXiv · 2306.07793

On the $\alpha$-index of minimally $k$-(edge-)connected graphs for small $k$

Jiayu Lou·Ligong Wang·Ming Yuan
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Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of vertex degrees of $G$. For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$. The largest eigenvalue of $A_\alpha(G)$ is called the $\alpha$-index or the $A_\alpha$-spectral radius of $G$. A graph is minimally $k$-(edge)-connected if it is $k$-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not $k$-(edge)-connected. In this paper, we characterize the minimally 2-edge-connected graphs and minimally 3-connected graph with given order having the maximum $\alpha$-index for $\alpha \in [\frac{1}{2},1)$, respectively.

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Jiayu Lou, Ligong Wang, Ming Yuan. 2023-06-13. On the $\alpha$-index of minimally $k$-(edge-)connected graphs for small $k$. https://arxiv.org/abs/2306.07793

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