The largest Laplacian eigenvalue of induced-$K_{1,r}$-free graphs

Let $G$ be a simple graph of maximum degree $d$, and let $\mu (G)$ denote the largest eigenvalue of its Laplacian matrix. For a fixed integer $k\ge 2$, Aharoni, Alon, and Berger (2016) asked whether every graph containing no induced copy of $K_{1,k}$ satisfies $\mu (G)\le (2-\frac{2}{k}+o(1))d$. We answer this question by proving the stronger sharp bound $$\mu (G)\le (2-\frac{2}{k})(d+1)\mathrm{.}$$ The proof combines a sign decomposition of a Laplacian Rayleigh vector with a weighted local Caro-Wei type inequality for independent sets.