On the Squared Distance Matrix of a Starlike Block Graph

Abstract: Let $D(G)$ be the distance matrix of a simple connected graph $G$. The Hadamard product $D(G)~\circ~ D(G)$ is called the squared distance matrix of $G$, and is denoted by $\Delta(G)$. A simple connected graph is called a starlike block graph if it has a central cut vertex, and each of its blocks is a complete graph. Let $\mathcal{S}(n_1, n_2, \ldots, n_b)$ be the starlike block graph with blocks $K_{n_1+1}, K_{n_2+1}, \ldots, K_{n_b+1}$ on $n=1 + \sum_{i=1}^b n_i$ vertices. In this article, we compute the determinant of $\Delta( \mathcal{S}(n_1, n_2, \ldots, n_b))$ and find its inverse as a rank-one perturbation of a positive semidefinite Laplacian-like matrix $\mathcal{L}$ with rank $n-1$. We also investigate the inertia of $\Delta( \mathcal{S}(n_1, n_2, \ldots, n_b))$. Furthermore, for a fixed value of $n$ and $b$, we determine the extremal graphs that uniquely attain the maximum and minimum spectral radius of the squared distance matrix for starlike block graphs on $n$ vertices and $b$ blocks.