A family of regular graphs of girth 5

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 $f (k)$, $k \ge 16$