Locally Homogeneous Graphs

For a connected graph G and point v of G, let G, be the subgraph induced by the points adjacent to v . This G is called locally Go if G, = Go for all points v of G. A graph is called locally homogeneous if it is locally Go for some Go. Recent work concerning local homogeneity has focused on two broad questions. First, for which Go does there exist a G that is locally Go? This has been settled for Go a cycle, linear forest, and certain trees [ 1, 21. Second is the question of characterization. For a specific graph Go, characterize all graphs G that are locally Go. For example, let K(n; t ) denote the complete multipartite graphK(n, n, . . . , n) , where there are t parts. When Go = K(n; t ) there is a unique G, namely G =K(n; t + 1) . A recent paper of Hall [4] classifies the graphs that are locally the Petersen graph. There are exactly three. In this Note, a large class of locally homogeneous graphs are obtained using groups. As a special case we characterize the graphs that are locally ncycles.