On singular distance matrices of unicyclic graphs

A unicyclic graph is a connected graph having exactly one cycle. It is known that the distance matrix $D(G)$ of a unicyclic graph $G$ is nonsingular if and only if the cycle in $G$ is of odd length. Motivated by the inverse formula for a nonsingular $D(G)$, in this paper, we establish an explicit Moore-Penrose inverse formula for the distance matrix of a unicyclic graph with even cycle. This formula is expressed as the sum of a symmetric Laplacian-like matrix and a rank one matrix. As consequences, we study the existence of an eigenvalue of $D(G)$ and deduce a known formula for the inertia of $D(G)$ when the cycle in $G$ is of even length.