A fan graph F_(m, n) is defined as the graph join K^__m + P_n, where K^__m is the empty graph on m nodes and P_n is the path graph on n nodes. The case m = 1 corresponds to the usual fan graphs, while m = 2 corresponds to the double fan, etc. Precomputed properties of fan graphs are implemented in the Wolfram Language as GraphData[{Fan, {m, n}}]. The (r, 2)-fan graph is isomorphic to the complete tripartite graph K_(1, 1, r), and the (r, 3)-fan graph to K_(1, 2, r). The fan graph F_(4, 1) is sometimes known as the gem graph.
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