Characterizing $A_\alpha$-minimizer graphs: given order and independence number

Jiaqi Zhang; Shuchao Li

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arXiv · 2508.09770

Characterizing $A_\alpha$-minimizer graphs: given order and independence number

Jiaqi Zhang·Shuchao Li

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Abstract

For a given graph $ G $, let $ A(G) $, $ Q(G) $, and $ D(G) $ denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of $ G $, respectively. The $ A_\alpha(G) $ matrix, proposed by Nikiforov, is defined as $ A_\alpha(G)=\alpha D(G)+(1 - \alpha)A(G) $, where $ \alpha\in[0,1] $. This matrix captures the gradual transition from $ A(G) $ to $ Q(G) $. Let $ \mathcal{G}_{n,\gamma} $ denote the family of all connected graphs with $ n $ vertices and independence number $ \gamma $. A graph in $ \mathcal{G}_{n,\gamma} $ is referred to as an $ A_\alpha $-minimizer graph if it achieves the minimum $ A_\alpha $ spectral radius. In this paper, we first demonstrate that the $ A_\alpha $-minimizer graph in $ \mathcal{G}_{n,\gamma} $ must be a tree when $ \gamma\geq\left\lceil\frac{n}{2}\right\rceil $, and we provide several characterizations of such $ A_\alpha $-minimizer graphs. We then specifically characterize the $ A_\alpha $-minimizer graphs for the case $ \gamma = \left\lceil\frac{n}{2}\right\rceil + 1 $ when $n\geq 9$. Furthermore, we obtain a structural characterization for the $ A_\alpha $-minimizer graph when $ \gamma=n - c $, where $ c\geq4 $ is an integer.

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Jiaqi Zhang, Shuchao Li. 2025-08-13. Characterizing $A_\alpha$-minimizer graphs: given order and independence number. https://arxiv.org/abs/2508.09770

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