Symmetric Y-graphs and H-graphs

Abstract

For any tree T with nodes all of degree 1 or 3, define an expansion of T as a cubic graph in which each node of T is replaced by a segment of n vertices, where n is an integer greater than 2, which are joined so that: 1. (1) a leaf of T is replaced by a cycle of length n, the edges joining vertices in an arithmetic sequence; 2. (2) a node of degree three is replaced by n disjoint vertices; 3. (3) an edge of T is replaced by n edges joining corresponding vertices in the appropriate segments. Frucht, Graver, and Watkins have shown that there are only six expansions of the tree consisting of a single edge, that is, the generalized Petersen graphs, that are symmetric. This paper shows that there are only six other expansions of trees that are symmetric. Four are Y-graphs, expansions of K 1.3, and two are H-graphs, expansions of the tree with two nodes of degree 3 and four nodes of degree 1. All these symmetric graphs were listed in Foster's Census in 1966.