Tight spectral conditions for the Hamiltonicity of $K_{1,r}$-free split graphs

The Hamiltonicity and related subjects of split graphs, and in particular $K_{1,r}$-free split graphs with $r\ge 3$ received much attention. Dai et al. [Discrete Math. 345 (2022) 112826] conjectured that every $(r-1)$-connected $K_{1,r}$-free split graph is Hamiltonian. They proved the case when $r=4$, and earlier Renjith and Sadagopan [Int. J. Found. Comput. Sci. 33 (2022) 1--32] proved the case when $r=3$. Recently, Liu, Song, Zhang and Lai [Discrete Math. 346 (2023) 113402] proved that a split graph is Hamiltonian if and only if it is fully cycle extendable. So for $r=3,4$ every $(r-1)$-connected $K_{1,r}$-free split graph is fully cycle extendable. We give tight spectral sufficient conditions for a $K_{1,r}$-free split graph to be Hamiltonian for $r=3,4$.