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# 学术预告-A Hoffman-type result on the limit points of the $A_{\alpha}$-spectral radius of graphs

Let $G$ be any simple graph, and let $A(G)$ denote its adjacency matrix. A. J. Hoffman determined the limit points of the spectral radius of the adjacency matrix of graphs less than $\sqrt{2+\sqrt{5}}$. In this paper, after giving an alternative version of Hoffman's results, we generalize them to Nikiforov's matrix $A_\alpha(G) = \alpha D(G)+(1-\alpha)A(G),$ where $\alpha \in [0,1)$ and $D(G)$ is the degree matrix of $G$. As a corollary, we retrieve the limit points of signless Laplacian spectral radius of graphs less than $2+{\tiny \frac{{\;}1{\;}}{3}\left((54 - 6\sqrt{33})^{\frac{{\;}1{\;}}{3}} + (54 + 6\sqrt{33})^{\frac{{\;} 1{\;}}{3}} \right)}$.

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