Editing Cubic graph (section)

==Symmetry==
In 1932, [[R. M. Foster|Ronald M. Foster]] began collecting examples of cubic [[symmetric graph]]s, forming the start of the [[Foster census]].<ref name="Ref_Foster">{{Citation|first1=R. M.|last1=Foster|title=Geometrical Circuits of Electrical Networks|journal=[[Transactions of the American Institute of Electrical Engineers]]|volume=51|pages=309–317|year=1932|doi=10.1109/T-AIEE.1932.5056068|issue=2|s2cid=51638449}}.</ref> Many well-known individual graphs are cubic and symmetric, including the [[Water, gas, and electricity|utility graph]], the [[Petersen graph]], the [[Heawood graph]], the [[Möbius–Kantor graph]], the [[Pappus graph]], the [[Desargues graph]], the [[Nauru graph]], the [[Coxeter graph]], the [[Tutte–Coxeter graph]], the [[Dyck graph]], the [[Foster graph]] and the [[Biggs–Smith graph]]. [[W. T. Tutte]] classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with each possible value of ''s'' from 1 to 5.<ref>{{Citation
 | doi = 10.4153/CJM-1959-057-2
 | last = Tutte
 | first = W. T.
 | journal = Can. J. Math.
 | pages = 621–624
 | title = On the symmetry of cubic graphs
 | url = http://cms.math.ca/cjm/v11/p621
 | volume = 11
 | year = 1959
 | s2cid = 124273238
 | doi-access = free
 | access-date = 2010-07-21
 | archive-date = 2011-07-16
 | archive-url = https://web.archive.org/web/20110716145555/http://cms.math.ca/cjm/v11/p621
 | url-status = dead
 }}.</ref>

[[Semi-symmetric graph|Semi-symmetric]] cubic graphs include the [[Gray graph]] (the smallest semi-symmetric cubic graph), the [[Ljubljana graph]], and the [[Tutte 12-cage]].

The [[Frucht graph]] is one of the five smallest cubic graphs without any symmetries:<ref>{{citation|last1=Bussemaker|first1=F. C.|last2=Cobeljic|first2=S.|last3=Cvetkovic|first3=D. M.|last4=Seidel|first4=J. J.|publisher=Dept. of Mathematics and Computing Science, Eindhoven University of Technology|series=EUT report|title=Computer investigation of cubic graphs|url=https://research.tue.nl/en/publications/computer-investigation-of-cubic-graphs|volume=76-WSK-01|year=1976}}</ref> it possesses only a single [[graph automorphism]], the identity automorphism.<ref>{{Citation | last1=Frucht | first1=R. | title=Graphs of degree three with a given abstract group | doi = 10.4153/CJM-1949-033-6 | mr=0032987 | year=1949 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=1 | issue=4 | pages=365–378| s2cid=124723321 | doi-access=free }}.</ref>