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

Abstract

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 $\ceil*{\frac{\mathbf{k}}{2}} \leq α(G) \leq \mathbf{k}-1$. 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 $K_{\alpha ,\mathbf{k}-\alpha }$.