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$\mathrm{.}Inthisarticle,weprovethatthemaximumspectralradius$\rho(G)$amongallgraphs$G$in$$\mathcal{B}$($\mathbf{k}$, \alpha)$, is uniquely attained for the complete bipartite graph $K_{\alpha, $\mathbf{k}$-\alpha}$.