A biplane is a 2-(k(k − 1)/2 + 1, k, 2) symmetric design. Only sixteen nontrivial biplanes are known: there are exactly nine biplanes with k < 11, at least five biplanes with k = 11, and at least two biplanes with k = 13. It is here shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete. This result further implies that there exists no 3-(57, 12, 2) des...

A subset of the vertex set of a graph G, S ⊆ V (G), is a (k, τ)-regular set if it induces a k-regular subgraph of G and every vertex not in the subset has τ neighbors in it. This paper is a contribution to the given problem of existence of (k, τ)-regular sets associated with all distinct eigenvalues of integral strongly regular graphs. The minimal idempotents of the Bose-Mesner algebra of stron...

We survey the relationships between two-weight linear [n, k] codes over GF(q), projective (n, k, /»„ /»,) sets in PG(£— \,q), and certain strongly regular graphs. We also describe and tabulate essentially all the known examples.

Definition of the strongly regular graphs on 77 and 100 vertices belonging to the simple groups M22 and HiS.

We consider directed strongly regular graphs de2ned in 1988 by Duval. All such graphs with n vertices, n6 20, having a vertex-transitive automorphism group, are determined with the aid of a computer. As a consequence, we prove the existence of directed strongly regular graphs for three feasible parameter sets listed by Duval. For one parameter set a computer-free proof of the nonexistence is pr...

The action of PGL(2,2m) on the set of exterior lines to a nonsingular conic in PG(2,2m) affords an association scheme, which was shown to be pseudocyclic in [H.D.L. Hollmann, Association schemes, Master thesis, Eindhoven University of Technology, 1982]. It was further conjectured in [H.D.L. Hollmann, Association schemes, Master thesis, Eindhoven University of Technology, 1982] that the orbital ...

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...

We present an algorithm for a construction of strongly regular families of trian-gulations for planar domains with a piecewise smooth boundary. Some additional properties of resulting triangulations are considered.

(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...