Events - 2023

Title: Nash-Williams Orientation for Infinite Graphs

Abstract: Nash-Williams proved in 1960 that an edge-connectivity of 2k is sufficient for a finite graph to admit a k-arc-connected orientation and conjectured that the same holds for infinite graphs. We show that the conjecture is true for locally finite graphs with countably many ends.

This is joint work with Max Pitz and Marcel Koloschin.

Title: The Container method for Enumerating Triangle-free graphs

Abstract: The container method is a tool for bounding the number of independent sets in graphs and can be generalized to hypergraphs. It utilizes the observation that independent sets are found in clusters or “containers” within the graph. One application of this method is to bound the number of finite objects with some forbidden substructure. For example, we can bound the number of graphs on N vertices that do not contain K₃ as a subgraph, that is, the number of Triangle-free graphs on N vertices. We will take a look at finding this bound using the container method.

Title: Wires, bits, and the cost of sorting

Abstract: How hard is it to sort a list of n integers? A basic course on algorithms says it's O(n log n) time, but what if the list is enormous - so big you would need to cover the surface of the moon just to store it?

Title: Polarization operators in superspace

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:00 pm.

Abstract: The classic coinvariant space is a graded representation of the symmetric group with deep connections to permutation statistics and Hall-Littlewood polynomials. Its generalization, the diagonal harmonics, has a rich connection to Macdonald polynomials and the q,t-Catalan numbers. In this talk, we consider the variant of the classical coinvariant story in the superspace. We briefly survey its connections and recent developments. We introduce polarization operators and discuss a new basis for its alternating component. We also discuss a folklore on cocharge and propose a basis for the super harmonics.

Title: Diagonal coefficients, graph invariants with the symmetries of Feynman integrals, and the proof of the c_2 completion conjecture

Abstract: In a scalar field theory the contribution of a Feynman diagram to the beta function of the theory, the Feynman period, can be written as an integral in terms of the (dual) Kirchhoff polynomial of the graph.

Title: Several Gyrafas-Sumner-type results for treewidth

Abstract: For a graph parameter which goes unbounded at both ends of a robust sparsity spectrum, one naturally asks if it is bounded anywhere in the middle. This is modelled after a well-known conjecture of Gyarfas and Sumner, asking the above question for the chromatic number as the parameter and the girth as the measure of sparsity. We look through this lens at the treewidth as the parameter and present a number of recent results answering the corresponding question in various settings. If time permits, we’ll try to sketch a proof as well as some directions for future work.

This is joint work with Bogdan Alecu, Maria Chudnovsky, Sophie Spirkl and partly with Tara Abrishami.

Title: Quantum Automorphism Groups of Trees

Abstract: We give a characterisation of quantum automorphism groups of trees. In particular, for every tree, we show how to iteratively construct its quantum automorphism group using free products and free wreath products. This can be considered a quantum version of Jordan’s theorem for the automorphism groups of trees. This is one of the first characterisations of quantum automorphism groups of a natural class of graphs with quantum symmetry. This talk is based on the following preprint: https://arxiv.org/abs/2311.04891

Title: Multiplicative Weight Update (MWU) Method for Solving Packing/Covering LPs

Abstract: I will present an overview of the MWU technique for solving Packing/Covering LPs easily and efficiently. The talk will be based on a combination of results from the following resources: (1) https://epubs.siam.org/doi/pdf/10.1137/1.9781611976014.11 (2) https://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf

Title: Distance-regular graphs that support a uniform structure

Abstract: Given a connected bipartite graph $G$, the adjacency matrix $A$ of $G$ can be decomposed as  $A=L+R$, where $L=L(x)$ and $R=R(x)$ are respectively the  lowering and the raising matrices with respect to a certain vertex $x$. The graph $G$ has a \textit{uniform structure} with respect to $x$ if the matrices $RL^2$, $LRL$, $L^2R$, and $L$ satisfy a certain linear dependency.

Title: The Tutte Polynomial, Bipartite Representations of Graphs, and Grid Walking 

Abstract:  The Tutte Polynomial has many equivalent definitions. It can be defined by a deletion-contraction relation with the terms determined by the sequence of contractions, deletions, loops, and isthmi. This definition is independent of edge order. Another definition relies on a fixed edge ordering and examines the edge activities over maximal spanning forests. There is a direct relationship between edge activity and deletion/contraction for a given edge ordering. Furthermore, the monomials of the Tutte polynomial can be interpreted as grid walks. This allows for an approach to the Tutte polynomial on hypergraphs by examining the grid walks of the bipartite representation of the graph.

Title: Kemeny’s constant and random walks on graphs

Abstract: Kemeny's constant is an interesting and useful quantifier of how well-connected the states of a Markov chain are. This comes to the forefront when the Markov chain in question is a random walk on a graph, in which case Kemeny's constant is interpreted as a measure of how `well-connected' the graph is. Though it was first introduced in the 1960s, interest in this concept has recently exploded. This talk will provide an introduction to Markov chains, an overview of the history of Kemeny’s constant, discussion of some applications, and a survey of recent results, with an emphasis on those that are extensions or generalizations of simple random walks on graphs, to complement Sooyeong’s talk from two weeks ago.

Title: Snake decompositions of lattice path matroids

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:00 pm.

Abstract: We study Ehrhart theory of lattice path matroid polytopes motivated by a conjecture by De Loera, Haws and Köppe. More specifically, we aim to understand the h*-vector of this family of matroid polytopes. To do so, we subdivide them into smaller matroid polytopes such that each piece is a snake, which are matroids such that their matroid polytope coincides with the order polytope of a fence poset.