The Fricke-Macbeath graph, also called the Macbeath graph, is the 1-skeleton of the Hurwitz map 
on the Fricke-Macbeath
curve, which has genus 7. It is a septic
graph with 72 vertices and 252 edges,
illustrated above in a number of embeddings. The
map has 168 triangular faces. Its automorphism
group has group order 1008 (Bokowski and Cuntz
2018).
The following cyclic construction organizes an edge list and two-half decomposition supplied by E. Pegg, Jr. (pers. comm., May 20-21, 2025). Take all subscripts
modulo 7 and write 
for the undirected edge
with endpoints 
and 
.
The letters are simply orbit labels. One 36-vertex half consists of a fixed vertex
and five seven-vertex families 
, 
, 
, 
, and 
, with one vertex in each family for every 
modulo 7. Simultaneously replacing every subscript 
by 
is an order-7 graph automorphism
of the half. For every 
, add the edges 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
. Take a disjoint isomorphic
primed copy on 
,
,
,
,
,
and 
.
Join the two halves, for every 
, by the edges 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
. The resulting graph is the Fricke-Macbeath graph.
The Fricke-Macbeath graph is implemented in the Wolfram Language as GraphData["FrickeMacbeathGraph"].
The local graph at every vertex is the cycle graph 
.
The graph is vertex-transitive, edge-transitive,
and arc-transitive, as well as Hamiltonian
and Hamilton-connected.
It has graph genus 7, girth 3, graph radius 5, graph diameter 5, chromatic number 5, edge chromatic number 7, clique number 3, and independence number 18. Its characteristic polynomial is
 |
of Genus 7: A Polyhedral Realization."
Art Discrete Appl. Math. 1, #P1.02, 2018.