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DIAMETER IN WALK GRAPHS

A walk W of length k is admissible if every two consecutive edges of W are distinct. If G is a graph, then its walk graph Wk(G) has vertex set identical with the set of admissible walks of length k in G. Two vertices are adjacent in Wk(G) if and only if one of the corresponding walks can be obtained from the other by deleting an edge from one end and adding an edge to the other end. We show that if the degree of every vertex in G is more than one, then Wk(G) is connected and we find bounds for the diameter of Wk(G).