The Tutte 12-cage, also called the Benson graph (Exoo and Jajcay 2008), is the unique 12-cage graph, equivalent to the generalized
hexagon 
and alternately called the generalized hexagon 
, as studied by Tits (1959). The
first implicit description as a graph seems to be by Benson
(1966), but it is most frequently called the Tutte 12-cage (Brouwer et al. 1989).
The Tutte 12-cage is implemented in the Wolfram Language as GraphData["Tutte12Cage"].
It has 126 vertices (each of vertex degree three), 189 edges, girth
12 (by definition), and diameter 6. It has graph
spectrum 
and has LCF notation [17, 27, 
, 
,
, 35, 
, 13, 
, 53, 
, 21, 57, 11, 
, 
, 59, ![-17]^7](/images/equations/Tutte12-Cage/Inline12.svg)
(Polster 1998). It is determined
by spectrum.
It is distance-regular with intersection array 
but is not distance-transitive.
A rectilinear drawing with 166 crossings was found by G. Exoo (pers. comm., May 12, 2019). QuickCross was used to produce a non-rectilinear drawing with 165 crossings (E. Weisstein, May 12, 2019). These give upper bounds of 166 and 165 for the rectilinear crossing number and graph crossing number, respectively.
It is the point-line Levi graph of the generalized hexagon 
with 63 points and 63 lines (A. E. Brouwer, pers. comm., Jun. 8, 2009).
It is the largest cubic distance-regular graph, but is not a cubic symmetric graph (Brouwer 1989). However, it is one of the five Iofinova-Ivanov graphs (i.e., bipartite cubic semisymmetric graphs whose automorphism groups preserve the bipartite parts and act primitively on each part). It is also the first in an infinite family of biprimitive graphs and was described by Biggs (1974, p. 164) in terms of the projective plane of order 9 equipped with a unitary form (Iofinova and Ivanov 1985).
In 1973, Balaban excised a tree consisting of two adjacent vertices and the twelve vertices within graph distance 2 to obtain the unique Balaban 11-cage.
The Tutte 12-cage has graph genus 17 (Metzger and Ulrigg 2025).
Incidence Graph of 
."
