2025/09/13 by Das, Joyentanuj, Mohanty, Sumit
#05C12 #05C50 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.10854
Let D(G) be the distance matrix of a simple connected graph G. The Hadamard product D(G)~∘~ D(G) is called the squared distance matrix of G, and is denoted by Δ(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 S(n1, n2, …, nb) be the starlike block graph with blocks Kn1+1, Kn2+1, …, Knb+1 on n=1 + ∑i=1b ni vertices. In this article, we compute the determinant of Δ( S(n1, n2, …, nb)) and find its inverse as a rank-one perturbation of a positive semidefinite Laplacian-like matrix L with rank n-1. We also investigate the inertia of Δ( S(n1, n2, …, nb)). 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.