2019/01/02 by Rubei, Elena, Ziani, Dario Villanis
#05C12 #05C22 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.00360
Given a positive-weighted simple connected graph with m vertices, labelled by the numbers 1,…,m, we can construct an m × m matrix whose entry (i,j), for any i,j∈\1,…,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 2n is not necessarily the n-hypercube graph. Finally we give a characterization of distance matrices of positive-weighted Petersen graphs.