Algebraic Graph Theory

Simeon Ball, Universitat Politécnica de Catalunya On subsets of a finite vector space in which every subset of basis size is a basis In this talk we consider sets of vectors S of the vector space Fq with the property that every subset of S of size k is a basis. The classical example of such a set is the following. Example (Normal Rational Curve) The set S = {(1, t, t, . . . , tk−1) | t ∈ Fq} ∪ {(0, . . . , 0, 1)}, is a set of size q +1. It is easily shown that S has the required property by checking that the k×k Vandermonde matrix formed by k vectors of S, has non-zero determinant. For q even and k = 3, one can add the vector (0, 1, 0) to S and obtain an example with q + 2 vectors. For these parameters, such a set of q + 2 vectors is called a hyperoval, and these have been studied extensively. There are many examples of hyperovals known which are not equivalent (up to change of basis and field automorphisms) to the example above. The only other known examples of size q + 1 is an example of size 10 in F9, due to Glynn, and an example in F2h due to Hirschfeld. The following conjecture exists in various areas of combinatorics. It is, known as the main conjecture for maximum distance separable codes, the representability of the uniform matroid in matroid theory, the embeddability of the complete design in design theory and Segre’s arcs problem in finite geometry. Conjecture A set of vectors S of the vector space Fq , k ≤ q − 1, with the property that every subset of S of size k is a basis, has size at most q + 1, unless q is even and k = 3 or k = q − 1, in which case it has size at most q + 2. I shall present a proof of the conjecture for q prime and discuss the non-prime case. I shall also explain how to then prove the following theorem, which is a generalisation Segre’s “oval is a conic” theorem in the case k = 3. Theorem If p ≥ k then a set S of q + 1 vectors of the vector space Fq , with the property that every subset of S of size k is a basis, is equivalent to the Normal Rational Curve example, where q = p. Ada Chan, York University Complex Hadamard Matrices and Strongly Regular Graphs An n × n matrix W is type II if WW (−)T = nI, where W (−) is the Schur inverse of W . If the entries of W have absolute value 1, then W is a complex Hadamard matrix. In this talk, we show that there are only five families of parameters for which the strongly regular graphs give complex Hadamard matrices. Using the Nomura algebras of these complex Hadamard matrices, we see that they do not arise from Diţă’s construction. Algebraic Graph Theory Banff International Research Station, 24–29 April 2011 Ameera Chowdhury, University of California San Diego On a conjecture of Frankl and Füredi Frankl and Füredi conjectured that if F ⊂ 2 is a non-trivial λ-intersecting family of size m, then the number of pairs {x, y} ∈ � X 2 � that are contained in some F ∈ F is at least � m 2 � [P. Frankl and Z. Füredi. A Sharpening of Fisher’s Inequality. Discrete Math., 90(1):103-107, 1991]. We verify this conjecture in some special cases, focusing especially on the case where F is additionally required to be k-uniform and λ is small. Sebastian Cioabǎ, University of Delaware On a conjecture of Brouwer regarding the connectivity of strongly regular graphs A (v, k,λ, μ)-strongly regular graph (SRG for short) is a finite undirected graph without loops or multiple edges such that (i) it has v vertices, (ii) it is regular of degree k, (iii) each edge is in λ triangles, (iv) any two nonadjacent points are joined by μ paths of length 2. The connectivity of a graph is the minimum number of vertices one has to remove in order to make it disconnected (or empty). In 1985, Brouwer and Mesner used Seidel’s characterization of strongly regular graphs with eigenvalues at least −2 to prove that the vertex-connectivity of any (v, k,λ, μ)-SRG equals its degree k. Also, they proved that the only disconnecting sets of size k are the neighborhoods N(x) of a vertex x of the graph. A natural question is: what is the minimum number of vertices whose removal will disconnect a (v, k,λ, μ)-SRG into non-singleton components? In 1996, Brouwer conjectured that this number is 2k − λ− 2. In this talk, I will report some progress on this problem. This is joint work with Kijung Kim and Jack Koolen (POSTECH, South Korea). Matt DeVos, Simon Fraser University Eigenvalues of (3, 6)-Fullerines An ordinary Fullerine is a cubic planar map where all faces have size 5 or 6. Such graphs may be realized as carbon molecules, and information about their spectrum has physical significance. Here we consider (3, 6)-Fullerines, where the faces may have sizes only 3 or 6. We show that these graphs may be realized as Cayley Sum graphs, and we use this to resolve a conjecture of Fowler, which asserts that all eigenvalues of such graphs come in pairs {x,−x} except for the four exceptional values {3,−1,−1,−1}. This is joint work with Goddyn, Mohar, and Samal. Algebraic Graph Theory Banff International Research Station, 24–29 April 2011 Graham Farr, Monash University Transforms, minors and generalised Tutte polynomials We introduce a family of transforms that extends graphand matroid-theoretic duality, and includes trinities and so on. Associated with each such transform are λ-minor operations, which extend deletion and contraction in graphs. We establish how the transforms interact with our generalised minors, extending the classical matroid-theoretic relationship between duality and minors: (M/e)∗ = M∗ \ e. We also introduce some generalised Tutte-Whitney polynomials based on these minor operations. Chris Godsil, University of Waterloo