Paley graphs

The orbital chromatic polynomial introduced by Cameron and Kayibi counts the number of proper $\lambda$-colorings of a graph modulo a group of symmetries. In this paper, we establish expansions for the orbital chromatic polynomial of the $n$-cycle for the group of rotations and its full automorphism group. Besides, we obtain a new proof of Fermat's Little Theorem.

In this article we introduce the concept of (p, α)-switching trees and use it to provide sufficient conditions on the abelian groups G and H for when Cay (G×H; S ∪B) is Hamiltondecomposable, given that Cay (G; S) is Hamilton-decomposable and B is a basis for H. Applications of this result to elementary abelian groups and Paley graphs are given.

A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of ‘Cartesian decompositions’ of the permuted set, relating them to certain ‘Cartesian systems of subgroups’. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.

Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over any group, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our axioms. However, for m>=3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective plan...

This paper concerns the general problem of classifying the finite deterministic automata that admit a synchronizing (or reset) word. (For our purposes it is irrelevant if the automata has initial or final states.) Our departure point is the study of the transition semigroup associated to the automaton, taking advantage of the enormous and very deep progresses made during the last decades on the theory of permutation groups, their geometry and their combinatorial structure. Let X be a finite set. We say that a primitive group G on X is synchronizing if G together with any non-invertible map on X generates a constant map. It is known (by some recent results proved by P. M. Neumann) that for some primitive groups G and for some singular transformations t of uniform kernel (that is, all blocks have the same number of elements), the semigroup G, t does not generate a constant map. Therefore the following concept is very natural: a primitive group G on X is said to be almost synchronizing if G together with any map of non-uniform kernel generates a constant map. In this paper we use two different methods to provide several infinite families of groups that are not synchronizing, but are almost synchronizing. The paper ends with a number of problems on synchronization likely to attract the attention of experts in computer science, combinatorics and geometry, groups and semigroups, linear algebra and matrix theory.

The core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which o...

We describe Forb{K_1,3, K_1,3}, the class of graphs G such that G and its complement G are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths or isolated vertices, and of the complements of these graphs. Considering the hypergraph H ^(3)(G) made of the 3-element subsets of the vertex set of a graph G on which G induces a clique or an independent subset, we deduce from above a description of the Boolean sum G+̇G' of two graphs G and G' giving the same hypergraph. We indicate the role of this latter description in a reconstruction problem of graphs up to complementation.

The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see 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 LLP to distinguish between cyclotomic schemes and s...

For a given permutation group G 6 Sym Ω, the class of maximal subgroups of Sym Ω that contain G is, in general, difficult to describe. In this paper we address the problem of describing a class of maximal subgroups of Sym Ω or Alt Ω that are full stabilisers of Cartesian decompositions of Ω and contain a given permutation group G with a transitive minimal normal subgroup M . We find that for a fixed ω ∈ Ω the set of G-invariant Cartesian decompositions of Ω is in one-to-one correspondence with the set of Gω -invariant Cartesian systems of subgroups of M . This result enables us to give a complete description of G-invariant Cartesian decompositions in the case where M is a non-abelian simple group.

A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented. Some consequences on the length of the synchronizing word are discussed.

Synchronization Suppose that you are given an automaton (whose structure you know) in an unknown state. You would like to put it into a known state, by applying a sequence of transitions to it. Of course this is not always possible! A reset word is a sequence of transitions which take the automaton from any state into a known state; in other words, the composition of the corresponding transitions is a constant mapping.

In collaboration with João Araújo and others, I have been thinking about relations between permutation groups and transformation semigroups. In particular, what properties of a permutation group G on a set Ω guarantee that, if f is any non-permutation on Ω (or perhaps any non-permutation of rank k), then the transformation semigroup 〈G, f〉 has specified properties. There is far too much to summarise in a short talk, but I will try to say a little about two topics: Synchronization: for example, what are the non-syncnronizing ranks of a permutation group, the ranks k for which 〈G, f〉 is nonsynchronizing for some element f of rank k? Regularity: for example, which permutation groups G have the property that, for any element f of rank k, f has a quasi-inverse in 〈G, f〉?

Considering uniform hypergraphs, we prove that for every non-negative integer h there exist two non-negative integers k and t with k ≤ t such that two h-uniform hypergraphs H and H ′ on the same set V of vertices, with V ≥ t, are equal up to complementation whenever H and H ′ are k-hypomorphic up to complementation. Let s(h) be the least integer k such that the conclusion above holds and let v(h) be the least t corresponding to k = s(h). We prove that s(h) = h + 2 ⌊log 2 h⌋. In the special case h = 2 or h = 2 + 1, we prove that v(h) ≤ s(h) + h. The values s(2) = 4 and v(2) = 6 were obtained in [9]. Let s(h) be the least integer k such that there is some t such that the the conclusion above holds and let v(h) be the least t corresponding to k = s(h). Theorem 2. (1) s(h) = h + 2 ⌊log 2 h⌋ ; (2) v(h) ≤ s(h) + h if h = 2 or h = 2 + 1.

A graph is cubelike if it is a Cayley graph for some elementary abelian 2-group Zn 2 . The core of a graph is its smallest subgraph to which it admits a homomorphism. More than ten years ago, Nešetřil and Šámal (On tension-continuous mappings. European J. Combin., 29(4):1025–1054, 2008) asked whether the core of a cubelike graph is cubelike, but since then very little progress has been made towards resolving the question. Here we investigate the structure of the core of a cubelike graph, deducing a variety of structural, spectral and group-theoretical properties that the core “inherits” from the host cubelike graph. These properties constrain the structure of the core quite severely — even if the core of a cubelike graph is not actually cubelike, it must bear a very close resemblance to a cubelike graph. Moreover we prove the much stronger result that not only are these properties inherited by the core of a cubelike graph, but also by the orbital graphs of the core. Even though the ...

Given group , the commuting graph of , is defined as the graph with vertex set , and two distinct vertices and are connected by an edge, whenever they commute, that is. In this paper we get some parameters of graph theory, as independent number and clique number for groups , .

Let q be a prime power, F q be the field of q elements, and k, m be positive integers. A bipartite graph G = G q (g, h, k, m) is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over F q , with two vertices (p 1 , p 2) ∈ P and [ l 1 , l 2 ] ∈ L being adjacent if and only if p 2 + l 2 = p k 1 l m 1. We prove that graphs G q (k, m) and G q (k , m) are isomorphic if and only if q = q and {gcd(k, q − 1), gcd(m, q − 1)} = {gcd(k , q − 1), gcd(m , q − 1)} as multisets. The proof is based on counting the number of complete bipartite subgraphs in the graphs.

The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.

Given group, the commuting graph of, is defined as the graph with vertex set, and two distinct vertices and are connected by an edge, whenever they commute, that is. In this paper we get some parameters of graph theory, as independent number and clique number for groups,. Keywords: independent number, clique number, generalized quaternion group

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics.

This paper concerns the general problem of classifying the finite deterministic automata that admit a synchronizing (or reset) word. (For our purposes it is irrelevant if the automata has initial or final states.) Our departure point is the study of the transition semigroup associated to the automaton, taking advantage of the enormous and very deep progresses made during the last decades on the theory of permutation groups, their geometry and their combinatorial structure.

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 edgeforwarding 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.

The results of several papers concerning theČerný conjecture are deduced as consequences of a simple idea that I call the averaging trick. This idea is implicitly used in the literature, but no attempt was made to formalize the proof scheme axiomatically. Instead, authors axiomatized classes of automata to which it applies. ⋆ The author gratefully acknowledges the support of NSERC

Given a finite field, one can form a directed graph  using the field elements as vertices and connecting two vertices if their difference lies in a fixed subgroup of the multiplicative group.
If −1 is contained in this fixed subgroup, then we obtain an undirected graph that is referred to as a generalized Paley graph. In this paper we study generalized Paley graphs whose clique and chromatic numbers coincide and link this theory to the study of the synchronization property in 1-dimensional primitive affine permutation groups.

A graph is called a Frobenius graph if it is a connected orbital graph of a Frobenius group. In this paper, we show first that almost all orbital regular graphs are Frobenius graphs. Then we give a description of Frobenius graphs in terms of a family of (usually smaller) Frobenius graphs which are Cayley graphs for elementary abelian groups. Finally, based on this description, we obtain a formula for calculating the edge-forwarding index of Frobenius graphs.

The synchronization property emerged from nite state automata and
transformation semigroup theory. Synchronizing permutation groups were introduced by Arnold and Steinberg to study the Cerny  Conjecture. In this thesis we study the synchronization property in affixne permutation groups of low-dimensions. J.E. Pin proved that one-dimensional affine groups are synchronizing. Hence our main results concern ane groups in dimension 2.
We used the characterization given by Neumann of synchronization using graph theory, which relies on the study of the equality between the clique number and the chromatic number of certain graphs  invariant under the actions of a group, called the suitability property. It turned out that some of such graphs for two-dimensional affine groups have an interesting geometry and are part of a widely studied class of graphs, the generalized Paley graphs.
Further, we used the properties of the theta-function defined by Lovasz connected to eigenvalues of a graph to obtain a necessary condition for the suitability in edge-transitive and vertex-transitive graphs. In this thesis we stated a criterion to decide if a generalized Paley graph is suitable. Then we used the tools referred to above and we presented conditions for two-dimensional affixne groups to be synchronizing.

Abstract. We study the minimal non-trivial subdegrees of finite primitive permutation groups that admit an embedding into a wreath product in product action, giving a connection with the same quantity for the primitive component. We discover that the primitive groups of twisted wreath type exhibit different (but interesting) behaviour from the other primitive types.