Murty [A generalization of the Hoffman–Singleton graph, Ars Combin. 7 (1979) 191–193.] constructed a family of $(pm + 2)$-\ud 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 $f (k)$, $k \ge 16$

The Heawood graph and $K_{3,3}$ have the property that all of their 2-factors are Hamilton circuits. We call such graphs 2-factor hamiltonian. We prove that if G is a k-regular bipartite 2-factor hamiltonian graph then either G is a circuit or k = 3. Furthermore, we construct an infinite family of cubic bipartite 2-factor hamiltonian graphs based on the Heawood graph and $K_{3,3}$ and conjecture that these are the only such graphs

We introduce a new method to describe tactical (de-)compositions of symmetric configurations via block (0,1)-matrices with constant row and column sum having circulant blocks. This method allows us to prove the existence of an infinite class of symmetric configurations of type $(2p^2)_{p+s }$ where p is any prime and s≤t is a positive integer such that t−1 is the greatest prime power with $t^2−t+1≤p$. In particular, we obtain a new configuration $98_{10}$

We show that a digraph which contains a directed 2-factor and has minimum in-degree and outdegree at least four has two non-isomorphic directed 2-factors. As a corollary, we deduce that every graph which contains a 2-factor and has minimum degree at least eight has two non-isomorphic 2-factors. In addition we construct: an infinite family of 3-diregular digraphs with the property that all their directed 2-factors are Hamilton cycles, an infinite family of 2-connected 4-regular graphs with the property that all their 2-factors are isomorphic, and an infinite family of cyclically 6-edge-connected cubic graphs with the property that all their 2-factors are Hamilton cycles.

Let G be a connected k-regular bipartite graph with bipartition V ðGÞ ¼ X [ Y and adjacency matrix A. We say G is det-extremal if per ðAÞ ¼ jdetðAÞj. Det-extremal k-regular bipartite graphs exist only for k ¼ 2 or 3. McCuaig has characterized the det-extremal 3-connected cubic bipartite graphs. We extend McCuaig's result by determining the structure of det-extremal cubic bipartite graphs of connectivity two. We use our results to determine which numbers can occur as orders of det-extremal connected cubic bipartite graphs, thus solving a problem due to H. Gropp.