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Archive : Discrete Mathematics

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Pat Devlin, Yale University 209Pat Devlin, Yale University Zoom<br> Title: "Buying Votes to (Probably) Win a Random Election" <br> <br> Abstract: In the low-stakes world of Duckburg politics, Scrooge McDuck is running for office, and he only has one opponent. Each morning leading up to the election, all the citizens of Duckburg wake up, they check social media, and they notice who their friends are planning to vote for. Then in the evening, everybody updates their profiles so that their new political opinion agrees with the majority of what their friends thought in the morning. [This updating process is called "majority dynamics".] <br> Senator McDuck isn't above greasing a few palms to buy some votes, but he is a notoriously tight-fisted miser who refuses to bribe more voters than he needs to. He doesn't know who's actually friends with whom, so ultimately he'll have to leave it up to chance. In this talk, we explore how many voters he needs to pay off in order for him to be 99% sure that he'll end up unanimously winning the election, and we'll also discuss how long it will take the citizens of Duckburg to reach some sort of consensus. Our main result is a central limit theorem for how many voters will favor McDuck after one day assuming the underlying graph is drawn from the Erd\H{o}s-R\'enyi model. This is joint work with Ross Berkowitz.5/19/2021 5:20:00 PM5/19/2021 6:20:00 PM
Nathan Lindzey, University of Colorado, Boulder206Nathan Lindzey, University of Colorado, BoulderZoom<br> Title: Ranks, Hooks, and Hamiltonian Cycles <br> <br> Abstract: We determine the ranks and p-ranks of a few interesting matrices that live in the symmetric group and perfect matching association schemes. We discuss what the ranks and p-ranks of these matrices have to do with the fine-grained computational complexity of counting Hamiltonian cycles. 5/12/2021 5:20:00 PM5/12/2021 6:20:00 PM
Ruth Luo, University of California, San Diego200Ruth Luo, University of California, San DiegoZoom<br> Title: Super-pancyclic graphs <br> Abstract: We say a hypergraph H is super-pancyclic if for every set of vertices A, there exists a Berge cycle of H whose base vertices are exactly the vertices in A. In this talk, we give necessary conditions for a hypergraph to be super-pancyclic and also prove that these conditions are sufficient in certain classes of hypergraphs. We will also discuss related problems for finding Berge cycles in hypergraphs. This is joint work with Misha Lavrov, Alexandr Kostochka, and Dara Zirlin. 5/5/2021 5:20:00 PM5/5/2021 6:20:00 PM
Mark Kempton, Brigham Young University196Mark Kempton, Brigham Young UniversityZoom<br> Title: Spectral Properties of Non-backtracking Random Walks <br> Abstract: Random walks on graphs play a critical role in modern graph theory and in many graph algorithms. Spectral properties of matrices associated with a graph are key in understanding random walks. A natural modification of a random walk is to forbid backtracking -- that is, the random walk cannot return to the vertex visited on the previous step. In this talk, I will discuss matrices that can be associated to a graph that describe non-backtracking walks, and will look at spectral properties of those matrices.4/28/2021 5:20:00 PM4/28/2021 6:20:00 PM

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