Some algebraic constructions of dense graphs of large girth and of large size (original) (raw)
Constructions of small regular bipartite graphs of girth 6
2011
In this article, some structures in the projective plane of order q are found which allow us to construct small kregular balanced bipartite graphs of girth 6 for all k ≤ q. When k = q, the order of these q-regular graphs is 2(q 2 −1); and when k ≤ q −1, the order of these k -regular graphs is 2(qk − 2). Moreover, the incidence matrix of a k -regular balanced bipartite graph of girth 6 having 2(qk − 2) vertices, where k is an integer and q is a prime power with 3 ≤ k ≤ q − 1, is provided. These graphs improve upon the best known upper bounds for the number of vertices in regular graphs of girth 6.
A New Series of Dense Graphs of High Girth
Bulletin of the American Mathematical Society, 1995
Let k ≥ 1 be an odd integer, t = ⌊ k+2 4 ⌋ , and q be a prime power. We construct a bipartite, q-regular, edge-transitive graph CD(k , q) of order v ≤ 2q k−t+1 and girth g ≥ k + 5. If e is the the number of edges of CD(k , q) , then e = Ω(v 1+ 1 k−t+1). These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g , g ≥ 5 , g = 11 , 12. For g ≥ 24 , this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for 5 ≤ g ≤ 23 , g = 11 , 12 , it improves on or ties existing bounds.
A construction of small regular bipartite graphs of girth 8
2009
Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq 2 − q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.
New Constructions of Bipartite Graphs on m, n Vertices with Many Edges and Without Small Cycles
Journal of Combinatorial Theory, Series B, 1994
For arbitrary odd prime power q and s ∈ (0, 1] such that q s is an integer, we construct a doubly-infinite series of (q 5 , q 3+s)-bipartite graphs which are biregular of degrees q s and q 2 and of girth 8. These graphs have the greatest number of edges among all known (n, m)-bipartite graphs with the same asymptotics of log n m, n → ∞. For s = 1/3, our graphs provide an explicit counterexample to a conjecture of Erdős which states that an (n, m)-bipartite graph with m = O(n 2/3) and girth at least 8 has O(n) edges. This conjecture was recently disproved by de Caen and Székely [2], who established the existence of a family of such graphs having n 1+1/57+o(1) edges. Our graphs have n 1+1/15 edges, and so come closer to the best known upper bound of O(n 1+1/9). 1. Introduction. All graphs we consider are simple. The order (size) of a graph G is the number of its vertices (edges). We denote the order of G by v = v(G), and the size of G
Large regular bipartite graphs
2014
A recent result of one of the authors says that every connected subcubic bipartite graph that is not isomorphic to the Heawood graph has at least one, and in fact a positive proportion of its eigenvalues in the interval [−1, 1]. We construct an infinite family of connected cubic bipartite graphs which have no eigenvalues in the open interval (−1, 1), thus showing that the interval [−1, 1] cannot be replaced by any smaller symmetric subinterval even when allowing any finite number of exceptions. Similar examples with vertices of larger degrees are considered and it is also shown that their eigenvalue distribution has somewhat unusual properties. By taking limits of these graphs, we obtain examples of infinite vertex-transitive r-regular graphs for every r ≥ 3, whose spectrum consists of points ±1 together with intervals [r − 2, r] and [−r, −r + 2]. These examples shed some light * Supported in part by NSERC PGS.
On the homogeneous algebraic graphs of large girth and their applications
Linear Algebra and its Applications, 2009
Families of finite graphs of large girth were introduced in classical extremal graph theory. One important theoretical result here is the upper bound on the maximal size of the graph with girth 2d established in even circuit theorem by Erdös. We consider some results on such algebraic graphs over any field. The upper bound on the dimension of variety of edges for algebraic graphs of girth 2d is established. Getting the lower bound, we use the family of bipartite graphs D(n, K) with n 2 over a field K, whose partition sets are two copies of the vector space K n . We consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory. Finally, we present a very short survey on applications of finite homogeneous algebraic graphs to coding theory and cryptography.
Graphs of Prescribed Girth and Bi-Degree
Journal of Combinatorial Theory, Series B, 1995
We say that a bipartite graph Γ(V 1 ∪ V 2 , E) has bi-degree r, s if every vertex from V 1 has degree r and every vertex from V 2 has degree s. Γ is called an (r, s, t)-graph if, additionally, the girth of Γ is 2t. For t > 3, very few examples of (r, s, t)-graphs were previously known. In this paper we give a recursive construction of (r, s, t)-graphs for all r, s, t ≥ 2, as well as an algebraic construction of such graphs for all r, s ≥ t ≥ 3.
2011
Let q be a prime power and r = 0, 1 . . . , q − 3. 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. Moreover, in this work two Latin squares of order q − 1 with entries belonging to {0, 1, . . . , q}, not necessarily the same, are defined to be quasi row-disjoint if and only if the Cartesian product of any two rows contains at most one pair (x, x) with x = 0. Using these quasi row-disjoint Latin squares we find (q − 1)-regular bipartite graphs of girth 6 with q 2 − q − 2 vertices in each partite set. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth 6.
A Construction of Small (q−1)(q-1)(q−1)-Regular Graphs of Girth 8
The Electronic Journal of Combinatorics, 2015
In this note we construct a new infinite family of (q−1)(q-1)(q−1)-regular graphs of girth 8 and order 2q(q−1)22q(q-1)^22q(q−1)2 for all prime powers qgeq16q\geq 16qgeq16, which are the smallest known so far whenever q−1q-1q−1 is not a prime power or a prime power plus one itself.