Limit points of $A_{\alpha }$-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 $\alpha$ close to zero. Precisely, we show that for $\alpha \in [0,1\mathrm{/}2)$ there exists a positive number ${\tau }_2(\alpha )>2$ such that any value $\lambda >{\tau }_2(\alpha )$ is an $A_{\alpha }$-limit point. We also determine the existence of other intervals for which all its points are $A_{\alpha }$-limit points.