The number of shortest paths in the (n, k)-star graph

We enumerate all of the shortest paths between any vertex υ and the identity vertex in an (n, k)-star graph by enumerating the minimum factorizations of υ in terms of the transpositions corresponding to edges in that graph. This result generalizes a previous one for the star graph, and can be applied to obtain the number of the shortest paths between a pair of vertices in some of the other similar structures. It also implies an algorithm to enumerate all such paths.

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