Overall data
Census of connected cubic vertextransitive graphs
On this page, we present some overall data extracted from our census of connected cubic vertextransitive graphs of order at most 1280. For basic definitions, see terminology.
Number of graphs of order at most n
The figure below shows the number of connected cubic
vertextransitive graphs of order at most n (these are the
black data points). Superimposed on this data (in gray) is
the graph of the function n>n^{2}/15, which
seems to be a close approximantion on the range
considered.
Number of graphs of order at most 1280 by type
The next table gives the number of connected cubic vertextransitive
graphs of order at most 1280, discriminating with respect
to whether the graphs are Cayley or not and with respect
to the number m of orbits of the automorphism group on the
arcs (in particular, for m=1, we have the arctransitive
graphs and for m=3, we have the GRRs).
 m=1
 m=2
 m=3
 Total

Cayley 
386 
11853 
97687 
109926 
nonCayley 
96 
1338 
0 
1434 
Total 
482 
13191 
97687 
111360 
The next figure compares the proportion of Cayley graphs
(in black), GRRs (with a dashed line) and dihedrants (in
gray) to the total number of connected cubic vertextransitive graphs of order at
most n.
Hamiltonian cycles
With the exception of the four wellknown exceptions (the Petersen and the Coxeter graphs, together with their truncations), all graphs in this census admit an Hamiltonian cycle.
Girth and diameter: extremal examples
Write n_{cay}(g) (respectively n_{vt}(g)) for the
smallest order of a cubic Cayley graph
(respectively cubic vertextransitive graph) of girth g. Using our census,
we can
compute the values of these two functions for g at most 16 and obtain the
following table.
g
 n_{cay}(g)
 n_{vt}(g)

3 
4 
4 
4 
6 
6 
5 
50 
10 
6 
14 
14 
7 
30 
26 
8 
42 
30 
9 
60 
60 
10 
96 
80 
11 
192 
192 
12 
162 
162 
13 
272 
272 
14 
406 
406 
15 
864 
620 
16 
1008 
1008 
Write m_{cay}(d) (respectively m_{vt}(d)) for the
largest order of a cubic Cayley graph
(respectively cubic vertextransitive graph) of diameter d. Using our
census, we can
compute the values of these two functions for d at most 8 and obtain some
lower
bounds for d at most 12.
d
 m_{cay}(g)
 m_{vt}(g)

2 
8 
10 
3 
14 
14 
4 
24 
30 
5 
60 
60 
6 
72 
82 
7 
168 
168 
8 
300 
300 
9 
>=506 
>=546 
10 
>=882 
>=1250 
11 
>=1250 
>=1250 
12 
>=1250 
>=1250 
Back to the main page.
by Primož Potočnik, Pablo Spiga and Gabriel
Verret, April 2014.