Inverse formula for distance matrices of gear graphs

TL;DR

This paper derives a new explicit formula for computing the Moore-Penrose inverse of the distance matrix specifically for gear graphs, extending known results for other star-like graphs.

Contribution

The paper provides the first explicit inverse formula for the distance matrix of gear graphs, a class not previously covered in existing literature.

Findings

Derived an explicit inverse formula for gear graph distance matrices

Extended the class of star-like graphs with known inverse formulas

Facilitated direct computation of inverses for gear graph distance matrices

Abstract

Distance matrices of some star like graphs are investigated in \cite{JAK}. These graphs are trees which are stars, wheel graphs, helm graphs and gear graphs. Except for gear graphs in the above list of star like graphs, there are precise formulas available in the literature to compute the inverse/Moore-Penrose inverse of their distance matrices. These formulas tell that if $D$ is the distance matrix of $G$, then $D^\dagger = -\frac{1}{2}L+uu'$, where $L$ is a Laplacian-like matrix which is positive semidefinite and all row sums equal to zero. The matrix $L$ and the vector $u$ depend only on the degree and number of vertices in $G$ and hence, can be written directly from $G$. The earliest formula obtained is for distance matrices of trees in Graham and Lov\'{a}sz \cite{GL}. In this paper, we obtain an elegant formula of this kind to compute the Moore-Penrose inverse of the distance matrix of a gear graph.