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Inverse formula for distance matrices of gear graphs

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 Lovász \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.