Paley Graphs Research Articles

Complete minors in pseudorandom graphs

Random graphs have the smallest complete minors among all graphs of given order and density. We show that, unlike random graphs, many well-known examples of pseudorandom graphs (such as Paley graphs) have very large minors. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 26–28, 2000

Complete minors in pseudorandom graphs

Random graphs have the smallest complete minors among all graphs of given order and density. We show that, unlike random graphs, many well-known examples of pseudorandom graphs (such as Paley graphs) have very large minors. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 26–28, 2000

Locally Paley Graphs

We conjecture that all locally Paley graphs are known. For locally Paley( p) with p < 100000 this is true.

Combinatorics of Monotone Computations

We demonstrate the criterion by super-polynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial t-designs. We then derive exponential lower bounds for clique-like graph functions of Tardos, thus establishing an exponential gap between the monotone real and non-monotone Boolean circuit complexities. Since we allow real gates, the criterion also implies corresponding lower bounds for the length of cutting planes proof in the propositional calculus.

Maximal cliques in the Paley graph of square order

Determining the clique number of the Paley graph of order q, q ≡ 1 (mod 4) a prime power, is a difficult problem. However, the work of Blokhuis implies that in the Paley graph of order q 2, where q is any odd prime power, the clique number is in fact q. In this paper we construct maximal cliques of size 1 2 (q + 1) or 1 2 (q + 3) , accordingly as q ≡ 1 (mod 4) or q ≡ 3 (mod 4), in the Paley graph of order q 2. It is believed that these are the largest maximal cliques which are not maximum. We also briefly discuss maximal cliques in some graphs naturally associated with the interior and exterior points of a conic in PG(2, q) for odd prime powers q.

On Edge Transitive Circulant Graphs

A classification is given for edge transitive circulant graphs whose complements are also edge transitive. In particular, it is shown that the only self-complementary symmetric circulant graphs are certain Paley graphs with a prime number of vertices. Similar results on digraphs are also obtained.

A note on graphs with a prescribed adjacency property

Let m and n be nonnegative integers and k be a positive integer. A graph G is said to have property P(m, n, k) if for any set of m + n distinct vertices of G there are at least k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertices. The problem that arises is that of characterising graphs having property P(m, n, k). This problem has been considered by several authors and a number of results have been obtained. In this paper, we establish a lower bound on the order of a graph having property P(m, n, k). Further, we show that all sufficiently large Paley graphs satisfy properties P(1, n, k) and P(n, 1, k).

The edge-forwarding index of orbital regular graphs

We define a graph as orbital regular if there is a subgroup of its automorphism group that acts regularly on the set of edges of the graph as well as on all its orbits of ordered pairs of distinct vertices of the graph. For these graphs there is an explicit formula for the edge-forwarding index, an important traffic parameter for routing in interconnection networks. Using the arithmetic properties of finite fields we construct infinite families of graphs with low edge-forwarding properties. In particular, the edge-forwarding index of Paley graphs is determined. A connection with the Waring problem over finite fields and the coset weight enumeration of certain cyclic codes is established.

On the adjacency properties of paley graphs

AbstractIn the application of graph theory to problems arising in network design, the requirements of the network can be expressed in terms of restrictions on the values of certain graph parameters such as connectivity, edge‐connectivity, diameter, and independence number. In this paper, we focus on networks whose requirements translate into adjacency restrictions on the graph representing the network. More specifically, a graph G is said to have property P(m,n,k) if for any set of m + n distinct vertices there are at least k other vertices, each of which is adjacent to the first m vertices but not adjacent to any of the latter n vertices. The problem that arises is that of characterizing graphs having property P(m,n,k). In this paper, we present properties of graphs satisfying the adjacency property. In particular, for q  1(mod 4), a prime power, the Paley graph Gq of order q is the graph whose vertices are elements of the finite field 𝔽q; two vertices are adjacent if and only if their difference is a quadratic residue. For any m, n, and k, we show that all sufficiently large Paley graphs satisfy P(m,n,k). © 1993 by John Wiley &amp; Sons, Inc.

Ramsey properties of orientations of graphs

AbstractA classical result of Rödl (in Graphs, Hypergraphs and Block Systems, 1976, pp. 211 to 220) says that for any acyclic digraph D there is a graph G = G(D) such that every acyclic orientation of G contains an induced copy of D. A recent result of Rödl and Winkler (SIAM J. Discrete Math., 2: 402–406, 1989) implies that there is such a graph G of order O(n3 (log n)2), where n = |D| is the order of D. Here we show by probabilistic means that almost every graph Gn of order [(4/9)n2n/2] has the property that every acyclic orientation of Gn contains an induced copy of every acyclic digraph on n vertices. We also show that the order of any such graph Gn has to be greater than (1/10)n2n/2. Thus our results are essentially best possible. Moreover, we show that there are Paley graphs of order O(n24n) having the property above. We also consider some problems raised by Cochand and Duchet (Irregularities of Partitions, Springer‐Verlag, Berlin, 1989, pp. 39–46.) on related topics. © 1993 John Wiley &amp; Sons, Inc.

Reconstructing graphs as subsumed graphs of hypergraphs, and some self-complementary triple systems

LetV be a set ofn elements. The set of allk-subsets ofV is denoted $$\left( {_k^V } \right)$$ . Ak-hypergraph G consists of avertex-set V(G) and anedgeset $$E(G) \subseteq \left( {_k^{V(G)} } \right)$$ , wherekÂ?2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvÂ?V(G) define a graphG v . The graphs {G v Â?vÂ?V(G)} aresubsumed byG. Each subsumed graphG v is a graph with vertex-setV(G) Â? v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachG v Â?H Â?v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG *, all of whose subsumed graphs are isomorphic toH, whereG andG * are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way.

Strong regularity and circulant graphs

Strongly regular pk-circulants, where p is a prime number, are characterised and a representation of Paley graphs of order p2 as metacirculants is given.

THE CLIQUE NUMBERS AND CHROMATIC NUMBERS OF CERTAIN PALEY GRAPHS

Abstract The problem of finding a formula for the clique number of the Paley graph G*(p) where p is a prime number such that p ≡ 1(mod 4) seems to be a hard one. These clique numbers are known for the caws 5 ⋚ p ⋚ 1601. In this paper we show that the clique number and chromatic number of the Paley graph G*(p2r) are both pr, where p is an odd prime and r is a natural number. Upper and lower bounds for the clique numbers and chromatic numbers of some related graphs are also determined.

CLIQUE NUMBERS OF PALEY GRAPHS

Abstract The clique number of the Paley graph G(q), where q, is a prime power with q ≡ 1 (mod 4) is known to be Jq, when q, is a square. When q, is a non-square the problem of finding a formula for the clique number seems to be a hard one although its value has been obtained by computer search for prime values of q, satisfying 5 ⋚ q, ⋚ 1601. In this paper we establish a lower bound asymptotically one half of the likely value.

Paley graphs satisfy all first‐order adjacency axioms

AbstractA graph satisfies Axiom n if, for any sequence of 2n of its points, there is another point adjacent to the first n and not to any of the last n. We show that, for each n, all sufficiently large Paley graphs satisfy Axiom n. From this we conclude at once that several properties of graphs are not first order, including self‐complementarity and regularity.