Geodesic
On the Non-Existence of a Type of Regular Graphs of Girth 5
Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 644-648

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ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that 1.1 Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph. 
Brown, William G. On the Non-Existence of a Type of Regular Graphs of Girth 5. Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 644-648. doi: 10.4153/CJM-1967-058-3
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[1] 1. Berge, C., Théorie des graphes et ses applications (Paris, 1958). Google Scholar

[2] 2. Brown, W. G., On Hamiltonian regular graphs of girth six, J. London Math. Soc., to appear. Google Scholar

[3] 3. Erdös, P. and Sachs, H., Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Univ. Halle, Math.-Nat., 12 (1963), 251–258. Google Scholar

[4] 4. Hoffman, A. J. and Singleton, R. R., On Moore graphs with diameters 2 and 3, IBM J. Res. Develop., 4 (1960), 497–504. Google Scholar

[5] 5. Mirsky, L., An introduction to linear algebra (Oxford, 1955). Google Scholar

[6] 6. Robertson, N., The smallest graph of girth 5 and valency 4, Bull. Amer. Math. Soc., 70 (1964), 824–825. Google Scholar

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