Limit points of Aα-matrices of graphs

We study limit points of the spectral radii of $A_{\alpha }$-matrices of graphs. Adapting a method used by J. B. Shearer in 1989, we prove a density property of $A_{\alpha }$-limit points of caterpillars for α close to zero. Precisely, we show that for $\alpha \in [0,\frac{1}{2})$ there exists a positive number ${\tau }_2\left(\alpha \right)>2$ such that any value $\lambda >{\tau }_2\left(\alpha \right)$ is an $A_{\alpha }$-limit point. We also determine other intervals whose numbers are all $A_{\alpha }$-limit points.