Spectral radius of $2$-dimensional simplicial complexes with given Betti number

In this paper we establish an asymptotic formula for the signless Laplacian spectral radius of a $2$-dimensional simplicial complex with given $2$-th Betti number. Furthermore, we characterize the $2$-dimensional simplicial complex that achieves the maximum signless Laplacian spectral radius among all-dimensional simplicial complex with the $2$-th Betti number equal to $1$ or $2$.