"The" H graph is the tree on 6 vertices illustrated above. It is implemented in the Wolfram Language as GraphData["HGraph"].
The term "H-graph" is also used to refer to a graph expansion with the 6-vertex H graph as its base (e.g., Horton and Bouwer 1991). There are exactly two graph expansions with H-graph base that are symmetric (Biggs 1993, p. 147).
graph | expansion | |
102 | Biggs-Smith graph | (17; 3, 5, 6, 7) |
204 | cubic symmetric graph | (34; 3, 5, 7, 11) |