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 $(pm + 2)$-\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 $f (k)$, $k \ge 16$

Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices

“…Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order q by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of (q − r)-regular bipartite graphs of girth 6 and q 2 − rq − 1 vertices in each partite set. q-regular bipartite graphs with q a prime power and 2(q 2 − 1) vertices of girth 6 have recently been obtained in [1]. We also improve this result for r = 1, finding (q − 1)-regular bipartite graphs of girth 6 with 2(q 2 − q − 2) vertices.…”

“…Furthermore, using quasi row-disjoint Latin squares we improve this result for r = 1 finding (q − 1)-regular bipartite graphs of girth 6 with q 2 −q −2 vertices in each partite set. Table 1 shows a comparison between the number of vertices of some known smallest graphs [1,20], and the graphs provided by the method proved in this work. We only consider degrees k in the interval [q − r, q + 1] in which the only prime power is q, and q − r is different from a prime power plus one.…”

“…Cages have been studied intensely since they were introduced by Tutte [21] in 1947. Erdős and Sachs [10] proved the existence of a graph for any value of the regularity k and the girth g, thus most of the work carried out has been focused on constructing a smallest one [1,3,5,8,11,12,18,16,17,23].…”

Regular graphs constructed from the classical generalized quadrangle Q(4, q)

“…This gives the bound c(k +1, 8) ≤ 2(1+q +q 2 +q 3 )−|B|. If k is close to q, then this upper bound is reasonably close to the lower bound (1). It is therefore of interest to construct large sets B with the given properties in Q (4, q).…”

“…Suppose that B is a set of some (but not all) points and lines of Q(4, q) (or any other generalized quadrangle of order q) such that every point and line not in B is incident with exactly q −k elements of B. Then the points and lines of Q(4, q) not in B define a graph with constant valency k +1; also if k is close to q, then (1) implies that the graph has girth 8. This gives the bound c(k +1, 8) ≤ 2(1+q +q 2 +q 3 )−|B|.…”