booklet - CUMC - Canadian Mathematical Society

minimum number of distinct eigenvalues of G. We will look at those graphs whichhave q(G) = 1 and q(G) = n. We will then discuss our conjecture for those graphs whichhave q(G) = n - 1 and the methods we are using to test this conjecture. Any knowledgeof graph theory needed for this talk will be briefly discussed at the beginning, makingthis topic accessible to all those with a basic course in linear algebra.A DISCRETE ANALOG OF COURANT’S THEOREMTHOMAS NGEigenvalues and eigenvectors of operators have been studied extensively due todirect applications in many areas of physics. Matrices, however, surface in variousseemingly unrelated fields such as Graph Theory. We provide a brief introduction toSpectral Graph Theory and some relations with Spectral Theory of Manifolds includinga combinatatorial version of Courant’s theorem on nodal domains of eigenfunctions.Required Background: Basic Linear AlgebraON THE NUMBER OF DIGITALLY CONVEX SETS IN TREESTIM PRESSEYLet G be a graph with vertex set V(G). We call a subset S of the vertex set of Gdigitally convex if for every v ∈ V(G), N[v] ⊆ N[S] implies v ∈ S. We prove sharpupper and lower bounds for the number of digitally convex sets of trees and showthat the number of digitally convex sets in a path on n vertices is exactly twice the nthFibonacci number.HODGE CONJECTURE IN SIMPLE TERMSTOMAS KOJARThe goal of this presentation is to try to de fine the objects that the Hodge conjectureis about. The Hodge conjecture proposes a deep connection between analysis, topology,and algebraic geometry. Very roughly it is saying that certain objects that are built viaanalysis ( differential forms) actually can be built via algebraic methods.DIAMOND HEIST: EXPLORING THE DOMINATION NUMBER OF A GRAPHVANESSA HALASAs a branch of mathematics, graph theory can often be considered too random andabstract. However, mathematicians have uncovered many patterns within graph theoryrelated to matrix structure, colouring, and classes of graphs. Domination, as arelatively new topic, did not gain popularity until the mid-1970’s, but nonetheless operatesin a similar and methodical way. In this short talk, I’ll provide a briefing intoconjectures of the domination number of a graph, the patterns that have been found, aswell as a look into the applications of the dominating set.48