Murty [A generalization of the Hoffman–Singleton graph, Ars Combin. 7 (1979) 191–193.] constructed a family of \$(pm + 2)\$- regular graphs of girth five and order \$2p^{2m}\$, where \$p \ge 5\$ is a prime, which includes the Hoffman–Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497–504]. This construction gives an upper bound for the least number \$f (k)\$ of vertices of a \$k\$-regular graph with girth 5. In this paper, we extend the Murty construction to \$k\$-regular graphs with girth 5, for each \$k\$. In particular, we obtain new upper bounds for \$f (k)\$, \$k \ge 16\$.

### A family of regular graphs of girth 5

#### Abstract

Murty [A generalization of the Hoffman–Singleton graph, Ars Combin. 7 (1979) 191–193.] constructed a family of \$(pm + 2)\$- regular graphs of girth five and order \$2p^{2m}\$, where \$p \ge 5\$ is a prime, which includes the Hoffman–Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497–504]. This construction gives an upper bound for the least number \$f (k)\$ of vertices of a \$k\$-regular graph with girth 5. In this paper, we extend the Murty construction to \$k\$-regular graphs with girth 5, for each \$k\$. In particular, we obtain new upper bounds for \$f (k)\$, \$k \ge 16\$.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11563/8937`
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