D. de Caen

An upper bound on the sum of squares of degrees in a graph

Abstract Let G be a simple graph with n vertices, e edges and vertex degrees d1, d2, …, dn. It is proved that d12 + … + dn2 ⩽ e(2e/(n − 1) + n − 2) when n ⩾ 2. This bound does not generalize to all sequences of positive integers. A comparison is made to another upper bound on d12 + … + dn2, due to Szekely et al. (1992). Our inequality follows from the positive semidefiniteness of a certain quadratic form in ( n 2 ) variables. We also apply the inequality to bounding the total number of triangles in a graph and its complement.

A lower bound on the probability of a union

Abstract A lower bound on the probability P ( A 1 ∪ ⋯ ∪ A m ) is presented, in terms of the P ( A i )s and P ( A i ∩ A j )s only. A comparison is made to a similar inequality due to Dawson and Sankoff (1967).

Algebraic multiplicity of the eigenvalues of a tournament matrix

Let F” denote the set of irreducible n X n tournament matrices. Here arc our main results: (1) For all n > 3, every matrix in K has at least three distinct eigenvalues; such a matrix has exactly three distinct eigenvalues if and only if it is a Hadamard tournament matrix. (2) For all n 2 3 there is a matrix in Y” having n distinct eigenvalues. (3) If cr, denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z, then ln /21-2 6. (4) If r,, is the minimum Perron value (i.e. spectral radius) of all matrices in r”, then 2 8.

A Family of Antipodal Distance-Regular Graphs Related to the Classical Preparata Codes

A new family of distance-regular graphs is constructed. They are antipodal 22t−1-fold covers of the complete graph on 22t vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs.There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matrices; and their underlying 3-class association scheme is formally dual to the scheme of a system of linked symmetric designs obtained from Kerdock sets of skew matrices in characteristic two.

Association Schemes Related to Kasami Codes and KerdockSets

Two new infinite series of imprimitive 5-class association schemes are constructed. The first series of schemes arises from forming, in a special manner, two edge-disjoint copies of the coset graph of a binary Kasami code (double error-correcting BCH code). The second series of schemes is formally dual to the first. The construction applies vector space duality to obtain a fission scheme of a subscheme of the Cameron-Seidel 3-class scheme of linked symmetric designs derived from Kerdock sets and quadratic forms over GF(2).

A Nonregular Analogue of Conference Graphs

We prove some results on graphs with three eigenvalues, not all integral; these are a natural generalization of the strongly regular conference graphs. We derive a Bruck?Ryser type condition and construct some (nonregular) examples.

On Dense Bipartite Graphs of Girth Eight and Upper Bounds for Certain Configurations in Planar Point-Line Systems

In earlier work we showed that ifG(m, n) is a bipartite graph with no 4-cycles or 6-cycles, and ifm

On the covering of t -sets with ( t +1)-sets: C(9,5,4) and C(10,6,5)

Abstract A (υ, k , t ) covering system is a pair ( X , B ) where X is a υ-set of points and B is a family of k -subsets, called blocks, of X such that every t -subset of X is contained in at least one block. The minimum possible number of blocks in a (υ, k , t ) covering system is denoted by C (υ, k , t ). It is proven that there are exactly three non-isomorphic systems giving C (9, 5, 4) = 30, and a unique system giving C (10, 6, 5) = 50.

Fissions of Classical Self-Dual Association Schemes

We describe fission schemes of most known classical self-dual association schemes, such as the Hamming scheme H(n, q) when q is a prime power. These fission schemes are themselves self-dual, with the exception of certain quadratic forms schemes in even characteristic.

Fissioned Triangular Schemes via the Cross-ratio

A construction of association schemes is presented; these are fission schemes of the triangular schemesT (n) where n=q+ 1 with q any prime power. The key observation is quite elementary, being that the natural action of PGL(2, q) on the 2-element subsets of the projective line PG(1, q) is generously transitive. In addition, some observations on the intersection parameters and fusion schemes of these association schemes are made.