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# WINCOM VIRTUAL COLLOQUIUM

We are pleased to continue this colloquium for the 2020-2021 academic year.

Each week we will be hosting a Zoom seminar by members of this network. If you would like to sign up for the email reminders for this colloquium, please join our email list below. The goal of the colloquium is to encourage community and collaboration within the network and the entire combinatorics community.

For Zoom issues, please email the organizer a reply to the colloquium announcement. For best results, log into Zoom in your browser and enter the meeting ID. This page is maintained by Dr. Erin Meger. Please reach out to her for any issues.

You can join this meeting via the Zoom link that will be sent out weekly. You can also find information about our seminar, and many other virtual talks at Combinatorics Lectures Online.

Slides from the talks are available where possible.

## SCHEDULE AND SPEAKER LIST

## Dr. Gabriela Araujo-Pardo, Instituto de Matemáticas, Universidad Nacional Autónoma de México

# Graph Theory and Finite Geometries

June 10, 2021, 9:00:00 p.m.

In this talk we mixed two different areas on combinatorics. Specifically, we take some results on finite geometries and combinatorial designs to solve classical problems on graph theory.

The first problem is related to extremal graphs and specifically to the construction of Moore graphs and cages. Here, we study this problem but on *Mixed Graphs*, where a Mixed regular graph is a *(z, r)*-graph, *z*-regular by arcs and *r*-regular by edges, a* (z, r; d)-mixed Moore graph* is a mixed graph with fixed diameter *d* and maximum order whereas a *[z, r; g]-mixed cage* is a mixed graph with fixed girth *g* and minimum order. In this talk we use the finite geometries and their incidence graphs to construct both mixed Moore graphs and cages.

The second problem studies complete colorations in circulant graphs. A complete *k*-vertex-coloring of a graph *G* is a vertex-coloring of *G *using *k* colors such that for every pair of colors there is at least two incident vertices in *G *colored with this pair of colors. The chromatic *χ(G) *and achromatic *α(G)* numbers of *G* are the smallest and the largest number of colors in a complete proper *k*-vertex-coloring of *G*, therefore *χ(G) ≤ α(G)*. If we coloring the edges, instead the vertices of a graph *G*, the analogous parameter is called the achromatic index , and denoted by *α _0 (G)*. In this talk, we determine the achromatic index for circulant graphs of order *q 2 + q + 1* using projective planes.

## Dr. Eimear Byrne, University College Dublin

# Cryptomorphisms of q-Matroids

June 3, 2021, 6:00:00 p.m.

A q-matroid is the q-analogue of a classical matroid. It comprises a rank function that satisfies particular axioms on the elements of the lattice of subspaces of a finite dimensional vector space. It turns out that there are several equivalent descriptions of a q-matroid. As with their classical counterparts, the flat, independence, circuit, closure, and hyperplane axioms (and more besides) all offer cryptomorphic descriptions of a q-matroid. We discuss the different axioms of q-matroids and highlight the difference between the classical theory and its q-analogue.

## Dr. Catherine Greenhill, University of New South Wales Sydney

# Mixing time of the switch Markov chain and stable degree sequences

May 14, 2021, 12:00:00 a.m.

The switch chain is a well-studied Markov chain which can be used to sample from the set of all graphs with a given degree sequence. Polynomial mixing time has been established for the switch chain under various conditions on the degree sequences. These conditions are related to notions of stability for degree sequences. I will discuss some results on this topic, including joint work with Jane Gao (Waterloo).

## Dr. Susama Agarwala, Johns Hopkins Applied Physics Lab

# Wilson loop diagrams and some observations on the product of Grassmann Necklace elements

May 7, 2021, 12:00:00 a.m.

In this talk, I discuss a class of variable valued matrices arising from the diagramatics of a physical theory (N=4 SYM). I use these diagrams to relate the interesting combinatorics of the theory to the combinatorics underlying the Grassmannians. Note: no physics is required for the understanding of this talk.

## Dr. Jane Gao, University of Waterloo

# The sandwich conjecture of random regular graphs

April 30, 2021, 12:00:00 a.m.

Random regular graphs are extensively studied, whose analysis typically involves highly nontrivial enumeration arguments, especially when the degree grows fast as the number of vertices grows. Kim and Vu made a conjecture in 2004 that the random regular graph can be well approximated by the random binomial random graphs, in the sense that it is possible to construct *G(n,d), G(n,p_1)* and *G(n,p_2)*, where *d~ np_1~ np_2*, in the same probability space so that with high probability *G(n,p_1) ⊆G(n,d) ⊆G(n,p_2)*. This is known as the sandwich conjecture. In this talk I will discuss some recent progress in this conjecture.

## Dr. Jo Ellis-Monaghan, University of Amsterdam/Korteweg-de Vries Institute for Mathematics

# Why party with lab scientists?

April 23, 2021, 12:00:00 a.m.

Why party with lab scientists? They are a lot of fun, and they have great parties. But more than that, they often have really interesting problems, problems that can potentially open new areas of mathematical investigation. There is a long history of this phenomenon in graph theory, where applications of immediate concern have led to new subfields. Examples include graph drawing and computer chip layout problems, random graph theory and modeling the internet, graph connectivity measures and ecological systems, etc. Currently, scientists are engineering self-assembling DNA molecules to serve emergent applications in biomolecular computing, nanoelectronics, biosensors, drug delivery systems, and organic synthesis. Often, the self-assembled objects, e.g. lattices or polyhedral skeletons, may be modeled as graphs. Thus, these new technologies in self-assembly are now generating challenging new design problems for which graph theory is a natural tool. We will explore these applications in DNA self-assembly together with the emergent theoretical directions that involve a synthesis of graph theory, knot theory, geometry, and computation.

## Dr. Kitty Meeks, University of Glasgow

# From decision to approximate counting

April 16, 2021, 12:00:00 a.m.

Decision problems – those in which the goal is to return an answer of “YES" or “NO" – are at the heart of the theory of computational complexity, and remain the most studied problems in the theoretical algorithms community. However, in many real-world applications it is not enough just to decide whether the problem under consideration admits a solution: we often want to find all solutions, or at least count (either exactly or approximately) their total number. It is clear that finding or counting all solutions is at least as computationally difficult as deciding whether there exists a single solution, and indeed in many cases it is strictly harder even to count approximately the number of solutions than it is to decide whether there exists at least one (assuming P is not equal NP).

In this talk I will discuss a restricted family of problems, in which we are interested in solutions of a given size: for example, solutions could be *k*-vertex subgraphs in a large host graph *G* that have some specific property (e.g. being connected), or size-*k* solutions to a database query. In this setting, although exact counting is strictly harder than decision (assuming standard assumptions in parameterised complexity), the methods typically used to separate approximate counting from decision break down. Indeed, I will describe two randomised approaches that, subject to some additional assumptions, allow efficient decision algorithms for problems of this form to be transformed into efficient algorithms to find or count all solutions.

This includes joint work with John Lapinskas (Bristol) and Holger Dell (Frankfurt).

## Dr. Puck Rombach, University of Vermont

# Expressing graphs as symmetric differences of cliques of the complete graph

April 2, 2021, 12:00:00 a.m.

Any finite simple graph *G = (V,E)* can be represented by a collection 𝒞of subsets of *V* such that *uv* ∈ *E* if and only if *u* and *v* appear together in an odd number of sets in 𝒞. We are interested in the minimum cardinality of such a collection. In this talk, we will discuss properties of this invariant and its close connection to the minimum rank problem. This talk will be accessible to students.

Joint work with Calum Buchanan and Christopher Purcell.

## Dr. Karen Meagher, University of Regina

# Erdős-Ko-Rado Theorems for Groups

March 26, 2021, 12:00:00 a.m.

The first half of this talk will be a gentle introduction to the

Erd**ő**s-Ko-Rado (EKR) Theorem. This is a theorem that determines the size and structure of the largest collection of intersecting sets. It has become a cornerstone of extremal set theory and has been extended to many other objects. I will show how this result can be proven using techniques from algebraic graph theory. In the second half of this talk I will give more details about extensions of the EKR theorem to permutations. Two permutations are intersecting if they both map some *i* to the same point (so σ and π are intersecting if and only if π^{-1}σ has a fixed point). In 1977, Deza and Frankl proved that the size of a set of intersecting permutations is at most (*n *- 1)!. It wasn't until 2003 that the structure of sets of intersecting permutations that meet this bound was determined. Since then, this area has developed greatly and I will give details about the recent results in this area.

## Dr. Karen Yeats, University of Waterloo

# The c_2 invariant of a graph via enumeration

March 20, 2021, 12:00:00 a.m.

The c_2 invariant is an arithmetic graph invariant that relates to Feynman integrals. I will explain the combinatorial set up of Feynman periods and the c_2 invariant. Then I will describe some results one can obtain from this perspective, with a focus on the prime 2.

Includes joint work with Oliver Schnetz and Simone Hu.

## Dr. Emily Sergel, Rutgers University

# Bar games and interpolation polynomials

March 12, 2021, 1:00:00 a.m.

The interpolation polynomials are a family of inhomogeneous symmetric polynomials characterized by simple vanishing properties. In 1996, Knop and Sahi showed that their top homogeneous components are Jack polynomials. For this reason these polynomials are sometimes called interpolation Jack polynomials, shifted Jack polynomials, or Knop-Sahi polynomials.

We prove Knop and Sahi's main conjecture from 1996, which asserts that, after a suitable normalization, the interpolation polynomials have positive integral coefficients. This result generalizes Macdonald's conjecture for Jack polynomials that was proved by Knop and Sahi in 1997. Moreover, we introduce a new combinatorial object called bar games and show that each interpolation polynomial equals a weighted sum of bar games.

This is joint work with Yusra Naqvi and Siddhartha Sahi.

## Dr. Christina Boucher, University of Florida

# Burrows Wheeler Transform meets the Colored de Bruijn Graph

March 5, 2021, 1:00:00 a.m.

In the past decade, there has been an effort to sequence and compare a large number of individual genomes of a given species, resulting in a large number of (reference) genomes of various species being made publicly available. For example, there is now public data for the 1,000 Genome Project, the 100K Genome Project, the 1001 Arabidopsis Genomes project, the Rice Genome Annotation Project, and the Bird 10,000 Genomes (B10K) Project. Key software, called genome assemblers, have been fundamental to the analysis of these datasets, and have enabled the creation of a large number of draft genomes. These methods take as input a set of sequence reads and typically build a de Bruijn graph from this data. Since the creation of the de Bruijn graph, many variations of the structure have been proposed and applied to biological data. One of the most prevalent variations has been the colored de Bruijn graph, which has been used to compare genome assemblies and sequence data. The application of this structure to large datasets have driven the creation of space- and time- efficient methods for building and storing the graph. In this talk, I will discuss one of these methods, which is based on Burrows Wheeler Transform (BWT). We will define the BWT of a colored de Bruijn graph and show experimentally how it can be used to analyze large biological datasets.