The Coxeter graph is a nonhamiltonian cubic symmetric graph on 28 vertices
and 42 edges which can be constructed as illustrated
above. It can also be constructed as the graph expansion
of 
with steps 1, 2, and 4, where 
is the claw
graph (Biggs 1993, p. 147). It is denoted 
in the Foster census of cubic
symmetric graphs.
It is distance-regular and distance-transitive with intersection array 
. It is also graceful
(E. Weisstein, Sep. 14, 2025).
A number of drawings are illustrated above (cf. Read and Wilson 1998, p. 162).
The Coxeter graph has orientable graph genus 3 and nonorientable graph genus 6, so its Euler genus is 6 (Mitchell 1995). In particular, it cannot be embedded in an orientable surface of genus 2. Such a graph embedding would have 12 faces by the Euler characteristic, and the graph's girth 7 and 42 edges would force every face to be a heptagon. Mitchell (1995) showed that an embedding of the Coxeter graph can have at most eight such faces.
The Coxeter graph has rectilinear crossing number 11, as established by G. Exoo around 1990 (G. Exoo, pers. comm., May 12, 2019). It is a cubic graph on 28 nodes with smallest possible graph crossing number of 11, making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2020).
As first shown by Bondy (1972), it is hypohamiltonian. It is also a platypus graph.
The Coxeter graph is determined by its spectrum 
(van Dam and Haemers 2002).
The bipartite double graph of the Coxeter graph is the cubic symmetric graph 
.
It is also a unit-distance graph, as illustrated in the above unit-distance embedding (Gerbracht 2008, pers. comm., Jan. 4, 2010).
The Coxeter graph is implemented in the Wolfram Language as GraphData["CoxeterGraph"].
If any edge is excised from the Coxeter graph, the resulting graph is the Hamilton-connected graph (left figure above) which is implemented in the Wolfram Language as GraphData["EdgeExcisedCoxeterGraph"]. The triangle-replaced Coxeter graph (right figure above) appears as an exceptional graph in conjectures about nonhamiltonian vertex-transitive graphs, H*-connected graphs, and Hamilton decompositions. It is implemented in the Wolfram Language as GraphData["TriangleReplacedCoxeterGraph"].
