Geodesic
Minimal Regular Graphs of Girths Eight and Twelve
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 1091-1094

Voir la notice de l'article provenant de la source Cambridge University Press

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal to Here the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems. 
Benson, Clark T. Minimal Regular Graphs of Girths Eight and Twelve. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 1091-1094. doi: 10.4153/CJM-1966-109-8
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[1] 1. Artin, Emil, Geometric algebra (New York, 1957). Google Scholar

[2] 2. Singleton, R., Regular graphs of even girth, Ph.D. Thesis, Princeton University (1963). Google Scholar

[3] 3. Tutte, W., Proc. Cambridge Philos. Soc. (1947). Google Scholar

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