On the spectral radius of bi-block graphs with given independence number $\alpha$

A connected graph is called a bi-block graph if each of its blocks is a complete bipartite graph. Let $\mathcal{B}(\mathbf{k}, \alpha)$ be the class of bi-block graph on $\mathbf{k}$ vertices with given independence number $\alpha$. It is easy to see that every bi-block graph is a bipartite graph. For a bipartite graph $G$ on $\mathbf{k}$ vertices, the independence number $\alpha(G)$ satisfies $\ceil*{\frac{\mathbf{k}}{2}} \leq \alpha(G) \leq \mathbf{k}-1$. In this article, we prove that the maximum spectral radius $\rho(G)$ among all graphs $G$ in $\mathcal{B}(\mathbf{k}, \alpha)$, is uniquely attained for the complete bipartite graph $K_{\alpha, \mathbf{k}-\alpha}$.