Paley Graphs Research Articles

A Generalization of Generalized Paley Graphs and New Lower Bounds for $R(3,q)$

Generalized Paley graphs are cyclic graphs constructed from quadratic or higher residues of finite fields. Using this type of cyclic graphs to study the lower bounds for classical Ramsey numbers, has high computing efficiency in both looking for parameter sets and computing clique numbers. We have found a new generalization of generalized Paley graphs, i.e. automorphism cyclic graphs, also having the same advantages. In this paper we study the properties of the parameter sets of automorphism cyclic graphs, and develop an algorithm to compute the order of the maximum independent set, based on which we get new lower bounds for $8$ classical Ramsey numbers: $R(3,22) \geq 131$, $R(3,23) \geq 137$, $R(3,25) \geq 154$, $R(3,28) \geq 173$, $R(3,29) \geq 184$, $R(3,30) \geq 190$, $R(3,31) \geq 199$, $R(3,32) \geq 214$. Furthermore, we also get $R(5,23) \geq 521$ based on $R(3,22) \geq 131$. These nine results above improve their corresponding best known lower bounds.

On generalised Paley graphs and their automorphism groups

The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the parameters for this to occur. We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in \cite{LLP} to distinguish between cyclotomic schemes and similar twisted versions of these schemes, in the context of homogeneous factorisations of complete graphs.

Maximal integral point sets in affine planes over finite fields

Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane F q 2 over a finite field F q , where the formally defined squared Euclidean distance of every pair of points is a square in F q . It turns out that integral point sets over F q can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case, integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over F q for q ≤ 47 . Furthermore, we give two series of maximal integral point sets and prove their maximality.

Block-diagonal semidefinite programming hierarchies for 0/1 programming

Lovász and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for 0/1 linear programming problems. We revisit these two constructions and propose two new, block-diagonal hierarchies, which are at least as strong as the Lovász–Schrijver hierarchy, but less costly to compute. We report experimental results for the stable set problem of Paley graphs.

Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

Let (K, + ,*) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, *) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 modulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter---Matthews presemifield and the Ding---Yuan variation of it, we recover a recent construction of skew Hadamard difference sets by Ding and Yuan [7]. On the other hand, applying this result to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the p-ranks of these pseudo-Paley graphs when q = 34, 36, 38, 310, 54, and 74. The p-rank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of Rene Peeters [17, p. 47] which says that the Paley graphs of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks.

Hardness of fully dense problems

In the past decade, there has been a stream of work in designing approximation schemes for dense instances of NP-Hard problems. These include the work of Arora, Karger and Karpinski from 1995 and that of Frieze and Kannan from 1996. We address the problem of proving hardness results for (fully) dense problems, which has been neglected despite the fruitful effort put in upper bounds. In this work, we prove hardness results of dense instances of a broad family of CSP problems, as well as a broad family of ranking problems which we refer to as CSP-Rank. Our techniques involve a construction of a pseudorandom hypergraph coloring, which generalizes the well-known Paley graph, recently used by Alon to prove hardness of feedback arc-set in tournaments.

Bounds for Ramsey numbers of complete graphs dropping an edge

Let Kn−e be a graph obtained from a complete graph of order n by dropping an edge, and let Gp be a Paley graph of order p. It is shown that if Gp contains no Kn−e, then r(Kn+1−e)≥2p+1. For example, G1493 contains no K13−e, so r(K14−e)≥2987, improving the old bound 2557. It is also shown that r(K2¯+G)≤4r(G,K2¯+G)−2, implying that r(Kn−e)≤4r(Kn−2,Kn−e)−2.

Cubic and quadruple Paley graphs with the [formula omitted]-e.c. property

A graph G is n-existentially closed or n-e.c. if for any two disjoint subsets A and B of vertices of G with | A ∪ B | = n , there is a vertex u ∉ A ∪ B that is adjacent to every vertex of A but not adjacent to any vertex of B. It is well-known that almost all graphs are n-e.c. However, few classes of n-e.c. graphs have been constructed. A good construction is the Paley graphs which are defined as follows. Let q ≡ 1 ( mod 4 ) be a prime power. The vertices of Paley graphs are the elements of the finite field F q . Two vertices a and b are adjacent if and only if their difference is a quadratic residue. Previous results established that Paley graphs are n-e.c. for sufficiently large q. By using higher order residues on finite fields we can generate other classes of graphs which we called cubic and quadruple Paley graphs. We show that cubic Paley graphs are n-e.c. whenever q ⩾ n 2 2 4 n - 2 and quadruple Paley graphs are n-e.c. whenever q ⩾ 9 n 2 6 2 n - 2 . We also investigate a similar adjacency property for quadruple Paley digraphs.

Some colouring problems for Paley graphs

The Paley graph P q , where q ≡ 1 ( mod 4 ) is a prime power, is the graph with vertices the elements of the finite field F q and an edge between x and y if and only if x - y is a non-zero square in F q . This paper gives new results on some colouring problems for Paley graphs and related discussion.

A Solution of a Problem of A. E. Brouwer

Let q=pe ? 1(mod 4) be a prime power, and let ?(q) be the Paley graph over the finite field $${\mathbb F}_q$$ . Denote by ?(q) the subgraph of ?(q) induced on the set of non-zero squares of $${\mathbb F}_q$$ . In this paper the full automorphism group of ?(q) is determined affirming the conjecture of Brouwer [Des. Codes Cryptograph. 21, 69--76 (2000)]. The proof combines spectral and Schur ring techniques.

Rigidity and separation indices of Paley graphs

It is shown that the ratio between separation and rigidity indices of graphs may be arbitrarily large. Paley graphs are such examples.

Pseudo‐random properties of self‐complementary symmetric graphs

AbstractThere are some results in the literature showing that Paley graphs behave in many ways like random graphs G(n, 1/2). In this paper, we extend these results to the other family of self‐complementary symmetric graphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 310–316, 2004

Distance-regular circulants

A classification of strongly regular Cayley graphs of cyclic group was independently achieved by Bridges and Mena (Ars Combin. 8 (1979) 143), Hughes, van Lint and Wilson (Announcement at the Seventh British Combinatorial Conference (1979)), and Ma (Discrete Math. 52 (1984) 75). Here we discuss their results in a more general setting of distance-regular Cayley graphs, and give the complete classification of distance-regular Cayley graphs of cyclic groups. In particular, we prove that a Cayley graph of a cyclic group is distance-regular if and only if it is isomorphic to a cycle, or a complete graph, or a complete multipartite graph, or a complete bipartite graph on a twice an odd number of vertices with a matching removed, or the Paley graph with a prime number of vertices.

A Prolific Construction of Strongly Regular Graphs with the $n$-e.c. Property

A graph is $n$-e.c.$\,$ ($n$-existentially closed) if for every pair of subsets $U$, $W$ of the vertex set $V$ of the graph such that $U\cap W=\emptyset$ and $|U|+|W|=n$, there is a vertex $v\in V-(U\cup W)$ such that all edges between $v$ and $U$ are present and no edges between $v$ and $W$ are present. A graph is strongly regular if it is a regular graph such that the number of vertices mutually adjacent to a pair of vertices $v_1,v_2\in V$ depends only on whether or not $\{v_1,v_2\}$ is an edge in the graph. The only strongly regular graphs that are known to be $n$-e.c. for large $n$ are the Paley graphs. Recently D. G. Fon-Der-Flaass has found prolific constructions of strongly regular graphs using affine designs. He notes that some of these constructions were also studied by Wallis. By taking the affine designs to be Hadamard designs obtained from Paley tournaments, we use probabilistic methods to show that many non-isomorphic strongly regular $n$-e.c. graphs of order $(q+1)^2$ exist whenever $q\geq 16 n^2 2^{2n}$ is a prime power such that $q\equiv 3\!\!\!\pmod{4}$.

All Self-Complementary Symmetric Graphs

In 1992, H. Zhang (J. Graph Theory 16, 1–5), using the classification of finite simple groups, gave an algebraic characterisation of self-complementary symmetric graphs. Yet, from this characterisation it does not follow whether such graphs, other than the well-known Paley graphs, exist. In this paper we give a full description of self-complementary symmetric graphs and their automorphism groups. In particular, we prove that apart from the Paley graphs there is another infinite family of self-complementary symmetric graphs and, in addition, one more graph not belonging to any of these families. We obtain this by investigating automorphism groups of graphs and applying classification results on primitive permutation groups of low rank. We prove also that the automorphism group of a self-complementary symmetric graph is permutation isomorphic to a subgroup of AΓL1(pr) with three exceptions, when it can be presented as a subgroup of AΓL2(pr).

On an adjacency property of almost all graphs

A graph is called n-existentially closed or n-e.c. if it satisfies the following adjacency property: for every n-element subset S of the vertices, and for every subset T of S, there is a vertex not in S which is joined to all of T and to none of S⧹ T. The unique countable random graph is known to be n-e.c. for all n. Equivalently, for any fixed n, almost all finite graphs are n-e.c. However, few examples of n-e.c. graphs are known other than large Paley graphs and examples of 2-e.c. graphs given in (Cacetta, et al., Ars Combin. 19 (1985) 287–294). An n-e.c. graph is critical if deleting any vertex leaves a graph which is not n-e.c. We classify the 1-e.c. critical graphs. We construct 2-e.c. critical graphs of each order ⩾9, and describe a 2-e.c.-preserving operation: replication of an edge. We also examine which of the well-known binary operations on graphs preserve n-e.c. for n=1,2,3.

The number of edges on generalizations of Paley graphs

Evans, Pulham, and Sheenan computed the number of complete 4‐subgraphs of Paley graphs by counting the number of edges of the subgraph containing only those nodes x for which x and x − 1 are quadratic residues. Here we obtain formulae for the number of edges of generalizations of these subgraphs using Gaussian hypergeometric series and elliptic curves. Such formulae are simple in several infinite families, including those studied by Evans, Pulham, and Sheenan.

Imprimitive Association Schemes of Low Ranks and Higmanian Graphs

A natural relationship between certain strongly regular graphs, called Higmanian graphs, and certain imprimitive association schemes (isa) of rank 4 is studied. A necessary and sufficient condition for Higmanian in terms of certain partition of their vertex sets is given, followed by a necessary easy-to-check condition in terms of their parameters. As a consequence, we show among others, that a Paley graph of perfect square order is Higmanian by giving an explicit construction. As a generalization, a rank 5 fission of the polar graph O5(3) is also included.

Hadamard Matrices and Strongly Regular Graphs with the $3$-e.c. Adjacency Property

A graph is $3$-e.c. if for every $3$-element subset $S$ of the vertices, and for every subset $T$ of $S$, there is a vertex not in $S$ which is joined to every vertex in $T$ and to no vertex in $S\setminus T$. Although almost all graphs are $3$-e.c., the only known examples of strongly regular $3$-e.c. graphs are Paley graphs with at least $29$ vertices. We construct a new infinite family of $3$-e.c. graphs, based on certain Hadamard matrices, that are strongly regular but not Paley graphs. Specifically, we show that Bush-type Hadamard matrices of order $16n^2$ give rise to strongly regular $3$-e.c. graphs, for each odd $n$ for which $4n$ is the order of a Hadamard matrix.

On graphs the neighbourhoods of whose vertices are strongly regular with

It is proved that an amply regular graph of diameter greater than 2 the neighbourhoods of whose vertices are strongly regular with is a Taylor graph. A description of the locally Paley graphs is obtained. Uniform extensions of the partial geometries are found.