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# Dyck graph

There is a unique cubic *symmetric* (i.e., both vertex- and
edge-transitive) graph on 32 vertices known as the Dyck graph.
## Construction

The Dyck graph is the graph that has as vertices the triangles
in the Shrikhande graph,
adjacent when they share an edge.
## Group

The group is (4x4):3:2:2 of order 192.
It is vertex- and edge-transitive, with vertex stabilizer Sym(3).
## Spectrum

The Dyck graph has spectrum
±3^{1}, (±√5)^{6}, ±1^{9},
and is the unique graph with this spectrum.
## Distribution diagram

Every vertex has a unique special antipode at distance 5,
but interchanging special antipodes is not a graph automorphism.
The graph is bipartite. The two components of the distance-2 graph
are copies of the Shrikhande graph.
The diameter is 5, the girth 6.

## References

Walther Dyck,
*Über Aufstellung und Untersuchung von Gruppe und Irrationalität
regulärer Riemann'scher Flächen*,
Math. Ann. **17** (1880) 473-509.
Wolfram, Dyck Graph.

Cubic symmetric xyz graphs.