Archive : Discrete Mathematics

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Jeremy L. Martin, University of Kansas221Jeremy L. Martin, University of Kansas<p>​<span class="wrap-text"><strong>Title:</strong> Simplicial Effective Resistance and Tree Enumeration<br><strong>Abstract:</strong> How many spanning trees does a connected graph have? Kirchhoff's classical Matrix-Tree Theorem provides a good general answer, expressing the number of spanning trees of a finite graph in terms of its Laplacian matrix. How does adding an edge to a graph affect the spanning tree count? Remarkably, one can answer this question by treating the graph as an electrical network and calculating the "effective resistance" between the endpoints of the new edge. This idea is useful for counting spanning trees of recursively defined classes of graphs; for example, Ehrenborg and van Willigenburg used effective resistance to give an elegant formula for Ferrers graphs. The Matrix-Tree Theorem extends from graphs to simplicial complexes (although the meaning of "count" has to be modified slightly to account for topological complications arising only in higher dimensions), and it turns out that the electrical method does too, as I will explain. Applications include a new proof of a result of Duval, Klivans and Martin on shifted complexes, and a proof of a conjecture of Aalipour and Duval on color-shifted complexes which generalize Ferrers graphs). This is joint work with Art Duval (U. Texas, El Paso) and Woong Kook and Kang-Ju Lee (Seoul National University).</span></p>12/9/2021 6:00:00 PM12/9/2021 7:00:00 PM
Felix Lazebnik, University of Delaware215Felix Lazebnik, University of Delaware<p><span class="wrap-text">Title: In Elliptic World<br> Abstract. All of us came across the terms like ellipse, elliptic integrals, elliptic functions, elliptic curves (which surprisingly are not ellipses) , theta functions, ℘-function, modular functions, automorphic functions, etc. In the recent 30 years many of these terms appeared in the discussion of the Wiles-Taylor proof of the Fermat Last Theorem. The names of those who created and worked in these areas include Diophantus, Fagnano, Euler, Gauss, Legendre, Abel, Jacobi, Eisenstein, Weierstrass , Kronecker, Mordell, Siegel, Weil, Faltings, Mazur, Elkies, Wiles, Taylor, and many others. Recently I tried to read more on the subjects, with a particular interest in understanding motivations for appearances of some related notions. Now I know \epsilon more, and I want to share it with you.</span><br></p>12/2/2021 6:00:00 PM12/2/2021 7:00:00 PM
Gilles Zemor, Universite de Bordeaux, France 216Gilles Zemor, Universite de Bordeaux, France <p><strong>​Title: </strong>Storage codes on graphs<br><strong>Abstract:</strong> Consider an assignment of bits to the vertices of a graph G=(V,E) with the property that the value of each vertex is a function of the values of its neighbors. A collection of such assignments is called a storage code of length |V| on G. The problem of maximizing the size of a storage code can be equivalently formulated as maximizing the probability of success in a guessing game on graphs. If G contains many cliques, it is easy to construct codes of rate close to 1, so a natural problem is to construct high-rate codes on triangle-free graphs, where constructing codes of rate >1/2 is a nontrivial task, with few known results. We show that triangle-free coset graphs of binary linear codes admit storage codes of rate >1/2, and surpass previously known rates. We also derive necessary conditions for such codes to have high rate, and even rate potentially close to one.​<br></p>11/18/2021 6:00:00 PM11/18/2021 7:00:00 PM
Li-An Chen, University of Delaware220Li-An Chen, University of Delaware<p><strong>​Title:</strong> Algorithms for computing the permutation resemblance of functions on finite groups<br></p><p><strong>Abstract:</strong> Given two functions f, g on a finite group (G;+), the resemblance of f to g is defined by the number of distinct images of the distinct difference f - g. For f : G --> G, the permutation resemblance of f is defined by the minimum resemblance of f to any permutation of G. In this talk, we introduce a linear programming algorithm which gives the permutation resemblance of any function. We shall also discuss an upper bound for specific functions obtained by analyzing a variation of this algorithm.​<br></p>11/11/2021 6:00:00 PM11/11/2021 7:00:00 PM

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