The Moore-Penrose Inverse of the Distance Matrix of a Helm Graph

2022/08/23 by Jeyaraman, I., Divyadevi, T., Azhagendran, R.
#05C12 #05C50 #15A09 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2208.10897

Abstract

In this paper, we give necessary and sufficient conditions for a real symmetric matrix, and in particular, for the distance matrix D(H_n) of a helm graph H_n to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86--104, 2021), for D(H_n) when n is even. Further, we derive a formula for the Moore-Penrose inverse of singular D(H_n) that is analogous to the formula for D(H_n)^-1. Precisely, if n is odd, we find a symmetric positive semidefinite Laplacian-like matrix L of order 2n-1 and a vector w∈ ℝ^2n-1 such that D(H_n)\ssymbol2 = -(1)/(2)L + (4)/(3(n-1))w\mathbfw^′, where the rank of L is 2n-3. We also investigate the inertia of D(H_n).

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