Graph of graphs

The pictures¹ in this page represent the space of trivalent graphs with various number of vertices. This space has a natural geometry, namely, two graphs are close if they differ by a few local mutations.

More precisely, we can give the space of graphs the structure of a graph itself. Let GG(n) be the graph whose vertices are trivalent graph with n=2k vertices and two vertices are connected by an edge if the associated graphs differ by a single Whitehead move. We are interested in the geometry of this graph. The graph GG(n) is always connected. The number of its vertices is of order n^(n/2) (see for example Bollobas) and its diameter is comparable to ( n log n ) (see Rafi-Tao). Here is depiction of GG(6): The graph GG(n) is natrually stratified by girth.(Recall that girth of a graph is the length of its shortest loop.) Let GG(n,g) be the subgraph of GG(n) consisting all graphs that have girth at least g. Here, we examine if these subgraphs are themselves connected.

Simple graphs

A graph is called simple if it has girth at least 3 (no loops or double edges). It was shown by Coco Zhang² that the subgraphs GG_simple(n) and GG_not_simple(n) are both connected. Hence to simplify our discussion, we concentrate on simple graphs only. We refer to GG_simple(n) = G(n) and to the subgraph of G(n) consisting of graphs of girth at least g by G(n,g).

Here are a few pictures of the graph of simple graphs. The vertices are colored by the girth of the associated graph as follows: At first each strada seem to be connected. On the right, we have the picture of the slice with the same color coding as before. But for larger values of n this is not true. For n = 18 and 20, the graphs of girth 3 and 4 are omitted to make the picture less crowded. For n=22, there are 90553 graphs of girth 5. Hence, we just depict the girth ≥ 6 subgraph, which is again disconnected. It seems like the "end" of G(n) is somewhat tree like. It is a question of McMullen whether this family of graphs is an expander family.

For larger n, the problem becomes computationally more difficult. Here is a table of the number of trivalent graphs of a given girth.³

	girth = 3	girth = 4	girth = 5	girth = 6	girth = 7
n = 4	1
n = 6	1	1
n = 8	3	2
n = 10	13	5	1
n = 12	63	20	2
n = 14	399	101	8	1
n = 16	3,268	743	48	1
n = 18	33,496	7,350	450	5
n = 20	412,943	91,763	5,751	32
n = 22	5,883,727	1,344,782	90,553	385
n = 24	94,159,721	22,160,335	1,612,905	7,573	1
n = 26	1,661,723,296	401,278,984	31,297,357	181,224	3
n = 28	31,954,666,517	7,885,687,604	652,159,389	4,624,480	21
n = 30	663,988,090,257	166,870,266,608	14,499,780,660	122,089,998	545
n = 32	14,814,445,040,728	3,781,101,495,300	342,646,718,608	3,328,899,586	30,368

1. 1. These pictures have been produce in collaboration with Roi Docampo and Jing Tao using Sage. Jing Tao's trip to the university of Toronto was funded by the GEAR network. ↩
2. 2. Coco Zhang is a high school junior at Branksome Hall. This work was done as part of the Math Mentorship Program at the University of Toronto during the winter of 2013.↩
3. 3. Source: http://oeis.org/A198303. See also here ↩