We survey recent results on constructions of difference sets and strongly regular Cayley graphs by using union of cyclotomic classes of finite fields. Several open problems are raised.

(otherwise we divide the vector by an appropriate scalar), so w.l.o.g. we have uj = 1 for a certain j ∈ {1, . . . , v}. The absolute value |(A~u)j| of the j-th component of A~u is at most ∑ i∼j |ui|; since the absolute values of all components of ~u are less than or equal to 1, we have ∑ i∼j |ui| ≤ k. On the other hand |(A~u)j| must be equal to |ρuj| = |ρ|, from which we obtain |ρ| ≤ k. If ρ = ...

In this paper, two results about the signless Laplacian spectral radius q(G) of a graph G of order n with maximum degree ∆ are proved. Let Bn = K2 + Kn denote a book, i.e., the graph Bn consists of n triangles sharing an edge. The results are the following: (1) Let 1 < k ≤ l < ∆ < n and G be a connected {Bk+1,K2,l+1}-free graph of order n with maximum degree ∆. Then q(G) ≤ 1 4 [ 3∆ + k − 2l + 1...

We consider the class ER(n, d, λ) of edge-regular graphs for some n > d > λ, i.e., graphs regular of degree d on n vertices, with each pair of adjacent vertices having λ common neighbors. It has previously been shown that for such graphs with λ > 0 we have n ≥ 3(d − λ) and much has been done to characterize such graphs when equality holds. Here we show that n ≥ 3(d − λ) + 1 if λ > 0 and d is od...

It is known that a linear two-weight code C over a finite field Fq corresponds both to a multiset in a projective space over Fq that meets every hyperplane in either a or b points for some integers a < b , and to a strongly regular graph whose vertices may be identified with the codewords of C . Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homo...

In an attempt to find a strongly regular graph of parameters (99, 14, 1, 2) or to disprove its existence, we studied its possible substructure and constructions.

We show that regular homogeneous two-weight Z pk -codes where p is odd and k > 2 with dual Hamming distance at least four do not exist. The proof relies on existence conditions for the strongly regular graph built on the cosets of the dual code.

We report on the complete computer search for a strongly regular graph with parameters (36,15,6,6) and chromatic number six. The result is that no such graph exists.

We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce ...

The Friendship theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let λ be any nonnegative integer and μ be any positive integer. Suppose each pair of friends have exactly λ common friends and each pair of strangers have exactly μ common friends in a pa...