Linear Algebra Techniques in Combinatorics & Graph Theory ( 11 w 5033 ) Jan . 30

S (in alphabetic order by speaker surname) Title: Geometric distance-regular graphs without 4-claws Speaker: Sejeong Bang, Seoul National University Abstract: A non-complete distance-regular graph Γ is called geometric if there exists a set C of Delsarte cliques such that each edge of Γ lies in a unique clique in C. In this talk, we determine the non-complete distance-regular graphs satisfying max{3, 3(a1 + 1)} < k < 4a1 + 10 − 6c2. To prove this result, we first show by considering non-existence of 4-claws that any non-complete distance-regular graph satisfying max{3, 8 3(a1 + 1)} < k < 4a1 + 10− 6c2 is a geometric distance-regular graph with smallest eigenvalue −3. Moreover, we classify the geometric distance-regular graphs with smallest eigenvalue −3. As an application, 7 feasible intersection arrays are ruled out. A non-complete distance-regular graph Γ is called geometric if there exists a set C of Delsarte cliques such that each edge of Γ lies in a unique clique in C. In this talk, we determine the non-complete distance-regular graphs satisfying max{3, 3(a1 + 1)} < k < 4a1 + 10 − 6c2. To prove this result, we first show by considering non-existence of 4-claws that any non-complete distance-regular graph satisfying max{3, 8 3(a1 + 1)} < k < 4a1 + 10− 6c2 is a geometric distance-regular graph with smallest eigenvalue −3. Moreover, we classify the geometric distance-regular graphs with smallest eigenvalue −3. As an application, 7 feasible intersection arrays are ruled out. Title: Maximal cocliques in Point-Hyperplane graphs Speaker: A. Blokhuis, Eindhoven University of Technology Abstract: We explain the proof of an Erdős-Ko-Rado type theorem for the Kneser graph on the pointhyperplane flags of a finite projective space. More precisely: If C is a set of point-hyperplane flags of PG(n − 1, q), such that for each pair (Pi, Hi), i = 1, 2 we have P1 ∈ H2 or P2 ∈ H1 (or both), then |C| ≤ 1 + 2q + 3q2 + · · · + (n− 1)qn−2. Related results and problems will also be discussed. We explain the proof of an Erdős-Ko-Rado type theorem for the Kneser graph on the pointhyperplane flags of a finite projective space. More precisely: If C is a set of point-hyperplane flags of PG(n − 1, q), such that for each pair (Pi, Hi), i = 1, 2 we have P1 ∈ H2 or P2 ∈ H1 (or both), then |C| ≤ 1 + 2q + 3q2 + · · · + (n− 1)qn−2. Related results and problems will also be discussed. Title: A Handful of Sparse Testing Problems Speaker: Charles J. Colbourn, Arizona State University Abstract: Consider the following questions: Consider the following questions: 1. Given a population of n items, of which at most t are defective, what is the smallest number of ‘pools’ to determine the defective items? 2. Given a faulty system of n components, of which at most t interact to create the fault, what is the smallest number of ‘tests’ to determine the faulty interaction? 3. Given an implementation of a boolean function of n variables, of which at most t are relevant, what is the smallest number of ‘queries’ to determine the function? 4. In a set of n users, each is given a copy of a piece of software in which specified bits carry a unique fingerprint. Given that a coalition of at most t users can collude to locate (as far as possible) the bit locations of the fingerprint, and change them arbitrarily, what is the fewest bits so that no user can be ‘framed’ by the coalition? 5. Given an n-dimensional signal, of which at most t coordinates are significant, what is the smallest number of ‘measurements’ to determine the signal?