Inverse of the Squared Distance Matrix of a Complete Multipartite Graph

2023/11/02 by Das, Joyentanuj, Mohanty, Sumit · 1 citation
#05C12 #05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.01069

Abstract

Let G be a connected graph on n vertices and d_ij be the length of the shortest path between vertices i and j in G. We set d_ii=0 for every vertex i in G. The squared distance matrix Δ(G) of G is the n× n matrix with (i,j)^th entry equal to 0 if i = j and equal to d_ij² if i ≠ j. For a given complete t-partite graph K_n1,n2,⋯,nt on n=∑_i=1^t n_i vertices, under some condition we find the inverse Δ(K_n1,n2,⋯,nt)^-1 as a rank-one perturbation of a symmetric Laplacian-like matrix L with \textuprank (L)=n-1. We also investigate the inertia of L.

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