On the spectral radius of block graphs having all their blocks of the same size

Let $\mathcal{B}(n,q)$ be the class of block graphs on $n$ vertices having all their blocks of the same size. We prove that if $G\in \mathcal{B}(n,q)$ has at most three pairwise adjacent cut vertices then the minimum spectral radius $\rho (G)$ is attained at a unique graph. In addition, we present a lower bound for $\rho (G)$ when $G\in \mathcal{B}(n,q)$.