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

Joyentanuj Das, Sumit Mohanty

Published 2020-04-09Version 1

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 every bi-block graph is a bipartite graph and 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 $K_{\alpha, \mathbf{k}-\alpha}$.