A characterization of distance matrices of weighted cubic graphs and Peterson graphs

TL;DR

This paper characterizes the distance matrices of positive-weighted hypercube and Petersen graphs, providing insights into their structure and distinguishing features within weighted graph theory.

Contribution

It offers new characterizations of distance matrices for weighted hypercube and Petersen graphs, expanding understanding of their structural properties.

Findings

Characterization of distance matrices of weighted hypercube graphs

Distinction between bipartite n-regular graphs and hypercube graphs

Characterization of distance matrices of weighted Petersen graphs

Abstract

Given a positive-weighted simple connected graph with $m$ vertices, labelled by the numbers $1,\ldots,m$, we can construct an $m \times m$ matrix whose entry $(i,j)$, for any $i,j\in\{1,\dots,m\}$, is the minimal weight of a path between $i$ and $j$, where the weight of a path is the sum of the weights of its edges. Such a matrix is called the distance matrix of the weighted graph. There is wide literature about distance matrices of weighted graphs. In this paper we characterize distance matrices of positive-weighted $n$-hypercube graphs. Moreover we show that a connected bipartite $n$-regular graph with order $2^n$ is not necessarily the $n$-hypercube graph. Finally we give a characterization of distance matrices of positive-weighted Petersen graphs.