# On distance matrices of graphs

## TL;DR

This paper extends the understanding of distance matrices in graphs by deriving formulas for their inverses, generalizing known results for trees and cactoid digraphs through the theory of Laplacian expressible matrices.

## Contribution

It introduces a novel approach to compute the inverse of distance matrices in complex graphs using Laplacian expressible matrices, generalizing previous results.

## Key findings

- Explicit formulas for the inverse of the distance matrix of weighted cactoid digraphs.
- Extension of Graham, Hoffman, and Hosoya's determinant formula to inverse matrices.
- Connection of the results to known formulas for trees and specific graph classes.

## Abstract

Distance well-defined graphs consist of connected undirected graphs, strongly connected directed graphs and strongly connected mixed graphs. Let $G$ be a distance well-defined graph, and let ${\sf D}(G)$ be the distance matrix of $G$. Graham, Hoffman and Hosoya [3] showed a very attractive theorem, expressing the determinant of ${\sf D}(G)$ explicitly as a function of blocks of $G$. In this paper, we study the inverse of ${\sf D}(G)$ and get an analogous theory, expressing the inverse of ${\sf D}(G)$ through the inverses of distance matrices of blocks of $G$ (see Theorem 3.3) by the theory of Laplacian expressible matrices which was first defined by the first author [9]. A weighted cactoid digraph is a strongly connected directed graph whose blocks are weighted directed cycles. As an application of above theory, we give the determinant and the inverse of the distance matrix of a weighted cactoid digraph, which imply Graham and Pollak's formula and the inverse of the distance matrix of a tree.