The Harries graph is one of the three 
-cage graphs, the other
two being the 
-cage
known as the Balaban 10-cage and the Harries-Wong
graph.
The Harries graph is Hamiltonian with 
Hamiltonian cycles. It has 678 distinct LCF
notations, four of which are order 5 (illustrated above) and 674 of which are
order 1. The order-5 LCF notations are ![[-35,-27,27,-23,15,-15,-9,-35,23,-27,27,9,15,-15]^5](/images/equations/HarriesGraph/Inline4.svg)
[![-35,9,15,-15,23,-27,27,-35,15,-15,-9,-27,27,-23]^5](/images/equations/HarriesGraph/Inline5.svg)
,
![[-33,-27,27,-21,-13,13,19,29,-29,33,21,-19,-13,13]^5](/images/equations/HarriesGraph/Inline6.svg)
,
and ![[-33,-27,27,9,-13,13,-21,29,-29,33,-13,13,-9,21]^5](/images/equations/HarriesGraph/Inline7.svg)
.
The plots above show the adjacency matrix, incidence matrix, and graph distance matrix for the Harries graph.
Notice that the Harries graph and Harries-Wong graph are cospectral graphs, meaning neither is determined by spectrum.
The following table summarizes properties of the Harries graph.
| automorphism group order | 120 |
| characteristic polynomial |  |
| chromatic number | 2 |
| claw-free | no |
| clique number | 2 |
| cospectral graph names | Harries-Wong graph |
| determined by spectrum | no |
| diameter | 6 |
| distance-regular graph | no |
| edge chromatic number | 3 |
| edge connectivity | 3 |
| edge count | 105 |
| Eulerian | no |
| girth | 10 |
| Hamiltonian | yes |
| Hamiltonian cycle count | 98304 |
| integral graph | no |
| independence number | 35 |
| perfect matching graph | no |
| planar | no |
| polyhedral graph | no |
| radius | 6 |
| regular | yes |
| square-free | yes |
| traceable | yes |
| triangle-free | yes |
| vertex connectivity | 3 |
| vertex count | 70 |
| weakly regular parameters |  |
