A characterization of distance matrices of weighted hypercube graphs and Petersen graphs

Given a positive-weighted simple connected graph with $m$ vertices, labelled by the numbers $1,ldots,m$, we can construct an $m imes m$ matrix whose entry $(i,j)$, for any $i,jin{1,dots,m}$, 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.