Suppose that n is even and a set of n 2 − 1 mutually orthogonal Latin squares of order n exists. Then we can construct a strongly regular graph with parameters (n, n 2 (n−1), n 2 ( 2 −1), n 2 ( 2 −1)), which is called a Latin square graph. In this paper, we give a sufficient condition of the Latin square graph for the existence of a projective plane of order n. For the existence of a Latin squa...

A partial difference set having parameters (n2, r(n − 1), n + r2 − 3r, r2 − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n2, r(n + 1),−n + r2 + 3r, r2 + r) is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory o...

We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected nonHamiltonian strongly regular graph on fewer than 99 vertices.

Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36,10,4,2), but there are 32548 non-isomorphic graphs with parameters (36,15,6,6). (The first assertion is a special case of a theorem of Shrikhande, while the second is the result of a computer search by McKay and Spence.) In the light of th...

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.