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 $k\geq 2$, Aharoni, Alon, and Berger (2016) asked whether every graph containing no induced copy of $K_{1,k}$ satisfies $\mu(G)\leq (2 - \frac{2}{k} + o(1)) d$. We answer this question by proving the stronger sharp bound $$\mu(G)\leq \left(2-\frac{2}{k}\right)(d+1).$$ The proof combines a sign decomposition of a Laplacian Rayleigh vector with a weighted local Caro-Wei type inequality for independent sets.