Maximization of the spectral radius of block graphs with a given dissociation number

A connected graph is called a block graph if each of its blocks is a complete graph. Let $\mathbf{B}\mathbf{l}(\mathbf{\text{k}},\phi )$ be the class of block graphs on $\mathbf{\text{k}}$ vertices with given dissociation number $\phi$. In this article, we have shown the existence and uniqueness of a block graph ${\mathbb{B}}_{\mathbf{\text{k}},\phi }$ in $\mathbf{B}\mathbf{l}(\mathbf{\text{k}},\phi )$ that maximizes the spectral radius $\rho (G)$ among all graphs $G$ in $\mathbf{B}\mathbf{l}(\mathbf{\text{k}},\phi )$. Furthermore, we also provide bounds on $\rho ({\mathbb{B}}_{\mathbf{\text{k}},\phi })$.