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A graph G of order n > 2 is pancyclic if G contains a cycle of length l for each integer l with 3 ≤ l ≤ n and it is called vertex-pancyclic if every vertex is contained in a cycle of length l for every 3 ≤ l ≤ n. A graph G of order n > 2 is Hamilton-connected if for any pair of distinct vertices u and v, there is a Hamilton u-v path, namely, there is a u-v path of length n − 1. The graph B(n) is a graph with the vertex setIn this paper, we show that the graph L(n) is vertex-pancyclic and Hamilton-connected whenever n ≥ 6.