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Given a positive-weighted simple connected graph with $m$ vertices, labelledby the numbers $1,\ldots,m$, we can construct an $m \times m$ matrix whoseentry $(i,j)$, for any $i,j\in\{1,\dots,m\}$, is the minimal weight of a pathbetween $i$ and $j$, where the weight of a path is the sum of the weights ofits edges. Such a matrix is called the distance matrix of the weighted graph.There is wide literature about distance matrices of weighted graphs. In thispaper we characterize distance matrices of positive-weighted $n$-hypercubegraphs. Moreover we show that a connected bipartite $n$-regular graph withorder $2^n$ is not necessarily the $n$-hypercube graph. Finally we give acharacterization of distance matrices of positive-weighted Petersen graphs.

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