Strongly regular Cayley graphs,skew Hadamard difference sets,and rationality of relative Gauss sums

In this paper, we give constructions of strongly regular Cayley graphs and skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields. Our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White (2002) [23] and several subfield examples into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.     

Some results on strongly regular graphs from unions of cyclotomic classes

In this paper, we construct some families of strongly regular graphs on finite fields by using unions of cyclotomic classes and index 2 Gauss sums. New infinite families of strongly regular graphs are found.     

Constructions of strongly regular Cayley graphs using index four Gauss sums

We give a construction of strongly regular Cayley graphs on finite fields $\mathbb{F}_{q}$ by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.     

Constructions of Strongly Regular Cayley Graphs Using Even Index Gauss Sums

In this paper, generalizing the result in [9], I construct strongly regular Cayley graphs by using union of cyclotomic classes of and Gauss sums of index w, where is even. In particular, we obtain three infinite families of strongly regular graphs with new parameters.     

Three-valued Gauss periods,circulant weighing matrices and association schemes

Gauss periods taking exactly two values are closely related to two-weight irreducible cyclic codes and strongly regular Cayley graphs. They have been extensively studied in the work of Schmidt and White and others. In this paper, we consider the question of when Gauss periods take exactly three rational values. We obtain numerical necessary conditions for Gauss periods to take exactly three rational values. We show that in certain cases, the necessary conditions obtained are also sufficient. We give numerous examples where the Gauss periods take exactly three values. Furthermore, we discuss connections between three-valued Gauss periods and combinatorial structures such as circulant weighing matrices and three-class association schemes.     

Lifting constructions of strongly regular Cayley graphs

We give two “lifting” constructions of strongly regular Cayley graphs. In the first construction we “lift” a cyclotomic strongly regular graph by using a subdifference set of the Singer difference sets. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs [7] and a construction of strongly regular Cayley graphs given in [15]. The two constructions are related in the following way: the second construction can be viewed as a recursive construction, and the strongly regular Cayley graphs obtained from the first construction can serve as starters for the second construction. We also obtain association schemes from the second construction.     

Two-Character Sets Arising from Gluings of Orbits

In this paper we construct two infinite families of transitive two-character sets and hence two infinite families of symmetric strongly regular graphs. We also construct infinite families of quasi-quadrics.     

Spectra of coronae

We introduce a new invariant, the coronal of a graph, and use it to compute the spectrum of the corona G°H of two graphs G and H. In particular, we show that this spectrum is completely determined by the spectra of G and H and the coronal of H. Previous work has computed the spectrum of a corona only in the case that H is regular. We then explicitly compute the coronals for several families of graphs, including regular graphs, complete n-partite graphs, and paths. Finally, we use the corona construction to generate many infinite families of pairs of cospectral graphs.     

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v₂+v₃,3v₁+v₂,3,3}-minihypers and some [15,4,9;3]-codes with B₂=0, J. Statist. Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.     

Infinite families of recursive formulas generating power moments of Kloosterman sums: O −(2n,2 r ) case

In this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the special orthogonal group SO ^?(2n, 2 ^r ). And we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of “Gauss sums” for the orthogonal groups O ^?(2n, 2 ^r ).     

Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets

In this paper, we give a new lifting construction of “hyperbolic” type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to m-ovoids and i-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.     

Values of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions,Gauss sums and trigonometric sums

Motivated by the classical work of Ramanujan and recent work of Berndt and Zaharescu, we establish certain infinite families of identities relating values of Dirichlet L-functions at positive integers to Gauss sums and trigonometric sums twisted with characters.     

Infinite class of new sign ambiguities

In 1934, two kinds of multiplicative relations, the norm and the Davenport-Hasse relations, between Gauss sums, were known. In 1964, H. Hasse conjectured that the norm and the Davenport-Hasse relations were the only multiplicative relations connecting Gauss sums over Fp. However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture. This counterexample was a new type of multiplicative relation, called a sign ambiguity, involving a ± sign not connected to elementary properties of Gauss sums. In this paper, we give an explicit product formula involving Gauss sums which generates an infinite class of new sign ambiguities, and we resolve the ambiguous sign by using Stickelberger?s theorem.     

On some exact sequences in the theory of cyclotomic fields

The purpose of this article is to show the existence of some exact sequences which relate the ideal class groups of cyclotomic fields to Gauss sums. These exact sequences imply the results of Hachimori, Ichimura and Beliaeva. Furthermore, we study the relations between the étale cohomology groups and some conjectures on cyclotomic fields.     

Vertex-magic labelings of regular graphs II

Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors to a regular graph of order n which possesses a strong VMTL, to produce a regular graph of the same order but larger size. In this paper, we exploit this construction method. We are able to show that for any r≥4, every r-regular graph of odd order n≤17 has a strong VMTL. We show how to produce strong labelings for some families of 2-regular graphs since these are used as the starting points of our construction. While even-order regular graphs are much harder to deal with, we introduce ‘mirror’ labelings which provide a suitable starting point from which the construction can proceed. We are able to show that several large classes of r-regular graphs of even order (including some Hamiltonian graphs) have VMTLs.     

On the existence of uncountably many matroidal families

A matroidal family is a set $\text{F}$ ≠ ? of connected finite graphs such that for every finite graph G the edge-sets of those subgraphs of G which are isomorphic to some element of $\text{F}$ are the circuits of a matroid on the edge-set of G. Simões-Pereira [5] shows the existence of four matroidal families and Andreae [1] shows the existence of a countably infinite series of matroidal families. In this paper we show that there exist uncountably many matroidal families. This is done by using an extension of Andreae's theorem, a construction theorem, and certain properties of regular graphs. Moreover we observe that all matroidal families so far known can be obtained in a unified way.     

Constructions of 1 1/2-designs from symplectic geometry over finite fields

In this paper, we construct some 1(1/2)-designs, which are also known as partial geometric designs, using totally isotropic subspaces of the symplectic space and generalized symplectic graphs. Furthermore, these 1(1/2)-designs yield six infinite families of directed strongly regular graphs.     

On a conjecture of Brouwer involving the connectivity of strongly regular graphs

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components.We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called Δ-spaces are counterexamples to Brouwer?s Conjecture. Using J.I. Hall?s characterization of finite reduced copolar spaces, we find that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field Fq (for any q prime power), and the strongly regular graphs constructed from the hyperbolic quadrics O⁺(2r,2) and from the elliptic quadrics O⁻(2r,2) over the field F₂, respectively, are counterexamples to Brouwer?s Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall?s characterization theorem for Δ-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of Δ-spaces and thus, yield other counterexamples to Brouwer?s Conjecture.We prove that Brouwer?s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles GQ(q,q) graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue −2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components.     

Explicit multiplicative relations between Gauss sums

H. Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguous sign in terms of the order of the ideal class of quadratic primes.