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

TL;DR

This paper characterizes the Moore-Penrose inverse of the distance matrix of helm graphs, providing explicit formulas and conditions, especially distinguishing between even and odd cases, and analyzing their inertia.

Contribution

It offers necessary and sufficient conditions for the Moore-Penrose inverse of helm graph distance matrices to be expressed as a sum of a Laplacian-like matrix and a rank-one matrix, with explicit formulas for both even and odd cases.

Findings

Explicit inverse formulas for even n

A new formula for the Moore-Penrose inverse when n is odd

Analysis of the inertia of the distance matrix

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 $\mathbf{w}\in \mathbb{R}^{2n-1}$ such that \begin{eqnarray*} D(H_n)\ssymbol{2} = -\frac{1}{2}L + \frac{4}{3(n-1)}\mathbf{w}\mathbf{w^{\prime}}, \end{eqnarray*} where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.