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 $\mathcal{S}(n_1,n_2,\dots ,n_b)$ be the starlike block graph with blocks $K_{n_1+1},K_{n_2+1},\dots ,K_{n_b+1}$ on $n=1+{\sum }_{i=1}^b{n}_i$ vertices. In this article, we compute the determinant of $\Delta (\mathcal{S}(n_1,n_2,\dots ,n_b))$ 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 $\Delta (\mathcal{S}(n_1,n_2,\dots ,n_b))$. 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.

arXiv comment19 pages