Bootstrap percolation is a process defined on a graph. In the first round, an initial set of infected vertices is selected. In subsequent rounds, uninfected vertices become infected if they are adjacent to at least r infected vertices. Once infected vertices remain infected. We are interested in the r-percolation number, the size of a minimum r-percolating set of a graph G, which is denoted m(G, r). Beginning with a simple lemma about 2-percolation in strongly regular graphs, we work our way towards determining the 2- percolation numbers of three infinite families of diameter 2, 2-connected graphs: conference graphs (which include Paley graphs), complementary prisms of Paley graphs, and those McKay-Miller-Sir ˇ a´n graphs which are built ˇ up from complementary prisms of Paley graphs. (Joint work with Rayan Ibrahim and Hudson LaFayette).

This talk investigates the chain of mathematical and historical events that led to the discovery of an intriguing and remarkable connection between two seemingly distinct areas of mathematics—a link that had gone unnoticed for over 1500 years. It encompasses number theory, algebra, combinatorics, and projective geometry, and includes contributions from mathematicians of all kinds, from the most distinguished to the relatively unknown.

Adrian Rice is the Dorothy and Muscoe Garnett Professor of Mathematics at Randolph-Macon College in Ashland, Virginia, where his research focuses on 19th- and early 20th-century British mathematics. In addition to papers on various aspects of the history of mathematics, his books include Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800–1945 (with Karen Hunger Parshall), Mathematics in Victorian Britain (with Raymond Flood and Robin Wilson), and Ada Lovelace: The Making of a Computer Scientist(with Christopher Hollings and Ursula Martin). He is a five-time recipient of awards for outstanding expository writing from the Mathematical Association of America, and in 2021 he was awarded the Catherine Richards Prize by the Institute of Mathematics and its Applications.

The generalized singular value decomposition (GSVD) helps us to compare and contrast the row spaces of two given matrices A and B. In particular, it identifies linear subspaces that are (a) exclusive to A, (b) exclusive to B, and (c) common to A and B. We will illustrate the GSVD through a novel application to low-rank matrix completion, show how the result can be highly sensitive to certain discrete parameters, and suggest a potential open problem related to enumerative geometry and Young’s lattice

The Dani-Mainkar construction assigns to each graph a 2-step nilpotent Lie algebra. Since two graphs are isomorphic if and only if the corresponding 2- step nilpotent Lie algebras are, one obtains a sort of dictionary between these two seemingly unrelated corners of mathematics. In this talk we describe a purely Lie-theoretic interpretation of independent sets of graphs that is compatible with the Dani-Mainkar construction. In particular, using graph theoretic-ideas as guiding principle, we are able to establish upper and lower bounds for the independence number of an arbitrary (not necessarily of DaniMainkar type) 2-step nilpotent Lie algebra. This is joint work with Daniele Grandini and the students from the 2024 NSF REU hosted by VCU.

A nonnegative n-by-n matrix is irreducible if its corresponding directed graph is strongly connected. The Perron-Frobenius theorem guarantees that an irreducible matrix has a unique (up to scaling) positive eigenvector with eigenvalue equal to the spectral radius. Multiplicatively topical maps are orderpreserving and homogeneous maps on the cone of nonnegative vectors in Rn that generalize nonnegative matrices. We’ll look at combinatorial and hypergraph conditions analogous to irreducibility for topical maps that guarantee existence and uniqueness of positive eigenvectors. We’ll also discuss examples of topical maps including max-algebra linear transformations and an example from XKCD.

A set of vertices of a graph is said to be in general position if no three vertices from the set lie on a common shortest path. Recently Klavzar, Rall and Yero generalized this notion by defining a set of vertices to be in general d-position if no three vertices from the set lie on a common shortest path of length at most d. We generalize this notion further by defining a set of vertices to be in k-general d-position if no k vertices of the set lie on a common shortest path of length at most d. The k-general d-position number of a graph is the largest cardinality of a k-general d-position set. We will discuss some bounds on the k-general d-position number of a graph and the computation of the kgeneral d-position number of finite paths and cycles. Along the way we will establish that the maximally even subsets of cycles, which were introduced in Clough and Douthett’s work on music theory, provide the largest possible kgeneral d-position sets in n-cycles. We also prove a formula for the k-general d-position number of certain thin finite grids, providing a partial answer to a question asked by Klavzar, Rall and Yero.

The Alternating Sign Matrix Polytope, ASMn is the convex hull of n × n matrices whose entries are 0, 1, and -1, whose non-zero entries alternate in sign, and whose row and column sums are 1. These polytopes were initially defined by Striker, and by Behrend and Knight. Brualdi and Dahl, as well as Lascoux, initiated the study of paths in the graph of the ASMn polytope. As with the earlier authors, we are looking for an upper bound on the distance in the graph from an ASM to the nearest permutation matrix. We will also investigate the combinatorial properties of faces of the ASM polytope.

Graphs have determinants. Their theory includes Sachs’ subgraphs and the Sachs’ determinant formula. We apply this to get a new product formula:

Let G be a graph with maximum critical independent set Ic. If X = Ic ∪ N(Ic), and Xbar = V(G) − X, then det(G) = det(G[X]) · det(G[Xbar].

Starting in dimensions 1 and 2 with the interval and the square, the binary partition polytope Bn of dimension n (> 2) is a polytope which “looks like” the lattice of faces of the binary partition polytope of dimension n − 1. The faces of Bn−1 and the vertices of Bn correspond to the (integer) partitions of 2 n into powers of 2. It is an open problem to devise a “good” way to compute the lattice operations (“join” and “meet”) on the binary partitions.

I explain and demonstrate (with video clips) my latest project, a pop-up book about how to visualize the tesseract and other n-dimensional cubes. I also discuss the mathematics that underlies some of the book’s pop-up mechanisms.

We introduce the problem of finding the size and structure of the largest intersecting family of paths in a graph. A family of sets (in this case, vertices of paths) is called intersecting if every pair of its members share an element. An intersecting family is called a star if some element is in every member of the family. The classic Erdős-Ko-Rado Theorem (1938, 1961) states, in the simplest case, that any intersecting family of r-subsets of {1, 2, ..., n}, when r ≤ n/2 , has size at most (n−1, r−1) and, when r < n/2, satisfies equality if and only if it is a star. In work with Neal Bushaw and former student James Danielsson, we prove analogous structural results for families of r-paths of a graph for several infinite classes of graphs.

Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. The pebbling number π(G) is the smallest t so that from any initial configuration of t pebbles it is possible, after a sequence of pebbling moves, to place a pebble on any given target vertex. Given any target or ’root’ vertex in the graph and any initial configuration of n pebbles on the graph, it is possible, after a possibly-empty series of pebbling moves, to reach a new configuration in which the designated root vertex has one or more pebbles. The first part of this talk considers the pebbling number of Kneser graphs, and gives positive evidence for the conjecture that every Kneser graph has pebbling number equal to its number of vertices and shows some new results and improved bounds. The second part of the talk will concentrate on target pebbling and show some techniques used to find extremal configurations and the target pebbling number in trees.

Imagine a game where two players take turns picking numbers from 1 to 100 with no repetition. The game ends when every number has either been chosen or is one away from a number that has been chosen, and the last player to make a move wins. Can you think of a winning strategy? In this talk we will show how to win this game, and then explore variants played on graphs and groups. In the case of graphs, the players take turns choosing vertices until the chosen vertices form a dominating set.

This is based on joint work with Sean Fiscus, Glenn Hurlbert, and Travis Pence.

A family of sets, each of size r, is called INTERSECTING when every pair of sets in the family contain a common element. Such a family is called a STAR when there is an element common to every set in the family. Erdos-Ko-Rado famously proved that when the universe of elements for our family contains at least 2r elements, then the largest intersecting families are always stars.

With Glenn Hurlbert and James Danielsson, we extend this result to a new setting. Here, we fix a graph G, and let our universe of sets be all those sets which are the vertex set of a length r path in G. For which graphs can we show that the largest intersecting families are stars? Are there graphs for which there are maximum size intersecting families which are stars as well as maximum size intersecting families which are non-stars? Are there any graphs where the ONLY maximum size intersecting families are non-stars?

We consider a variation of the pursuit-evasion game Cops and Robber in which the robber is invisible to the cop side unless outside the neighbourhood of at least one of the cops. Our parameter of interest, the hyperopic copnumber, is analogous to the copnumber; i.e. the hyperopic copnumber of a graph G is the minimum number of cops that suffice to guarantee a win on G. We present a variety of results, including a characterization of the graphs with hyperopic copnumber 1. We also characterize those graphs with largest possible hyperopic copnumber, and consider graphs in terms of diameter and planarity

A family of spanning trees of the complete graph on n vertices Kn is t-intersecting if any two members have a forest on t edges in common. We prove an Erdős-Ko-Rado result for t-intersecting families of spanning trees of Kn. In particular, we show there exists a constant C > 0 such that for all n ≥ C(log n)t the largest t-intersecting families of spanning trees of Kn are the families consisting of all spanning trees that contain a fixed set of t disjoint edges (as well as the stars on n vertices for t = 1). The proof uses the spread approximation technique in conjunction with the Lopsided Lovász Local Lemma.

This is joint work with Peter Frankl, Glenn Hurlbert, Ferdinand Ihringer, Andrey Kupavskii, Karen Meagher, and Venkata Raghu Tej Pantangi

A proper k-coloring of a graph G assigns to each vertex v a color α(v), with α(v) ∈ {1, . . . , k} such that α(v) 6= α(w) for every edge vw. (A list coloring is similar, except that distinct vertices may have distinct lists of allowable colors.)

A recoloring step in a graph G for a coloring α recolors some vertex v with a color allowable for v that is not used by α on any neighbor of v, yielding a new proper coloring. Given proper colorings α and β of G, we ask questions like: Can we transform α to β by a sequence of recoloring steps? And: Over all α and β, what is the longest that a shortest sequence from α to β can be?

In this talk we survey results on reconfiguration of colorings and list colorings. We end with a few conjectures.

Here we merge the two fields of Cops and Robbers and Graph Pebbling to introduce the new topic of Cops and Robbers Pebbling. Both paradigms can be described by moving tokens (the cops) along the edges of a graph to capture a special token (the robber). In Cops and Robbers, all tokens move freely, whereas, in Graph Pebbling, some of the cop tokens disappear with movement while the robber is stationary.

In Cops and Robbers Pebbling, some of the cop tokens disappear with movement, while the robber moves freely. We define the cop pebbling number of a graph to be the minimum number of cops necessary to capture the robber in this context, and present upper and lower bounds and exact values, some involving various domination parameters, for an array of graph classes. We also offer several interesting problems and conjectures.

This is joint work with Nancy Clarke and former VCU Masters student Josh Forkin.

A plumbed manifold is a type of topological space that is defined by gluing together disk bundles. The goal of this talk is to construct infinite families of invariants of such plumbed 3-manifolds.

Each invariant takes the form of an infinite series and is produced from a variety of combinatorial ingredients: (1) vertex weighted trees, (2) a root system and its Weyl group, (3) formal “admissible” series, and (4) spin-c structures. Ingredient #3 can also be interpreted as the solution to a strictly combinatorial puzzle defined on the root system. We’ll also discuss how these families of invariants generalize the “Z-hat” series defined by Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park and Ri.

This is joint work with N. Tarasca.

We assign a color (red or blue) to each vertex of a graph, called a red-blue coloring, and it is permissible for adjacent vertices to get the same color. A graph is called a closed neighborhood balanced graph (CNBC) if it has a red-blue coloring in which for each vertex v there are an equal number of red and blue vertices in v’s closed neighborhood. Analogously when open neighborhoods are considered the resulting graphs are called neighborhood balanced graphs (NBC). Both NBC and CNBC graphs can model situations where it is desirable to distribute two types of resources to vertices in a graph so that each vertex has a balance of resources among its neighbors.

In this talk we discuss a variety of examples and results about CNBC and NBC graphs. We show that the classes of NBC and CNBC graphs are not hereditary, and in fact, every graph is an induced subgraph of an NBC graph and of a CNBC graph. While red-blue colorings of NBC and CNBC graphs are locally balanced, we show that globally they may contain arbitrarily more red than blue vertices. We also consider trees and characterize those trees that are CNBC graphs.

The convergence of machine learning and optical character recognition (OCR) has sparked interest in the Ge’ez writing system, which presents unique challenges for achieving accurate and efficient text recognition. This ancient script, primarily utilized in the Horn of Africa, includes a wealth of handwritten secular and religious documents that necessitate digitization and character recognition. In this presentation, we will explore how modeling Ge’ez characters as graphs allows for the application of graph comparison methods to identify similarities and differences among the characters within the writing system. Specifically, we will use random walk kernels and the direct product of graphs to examine the relationships among various families of Ge’ez characters.

A Paley graph Pₚ is defined on the finite field with p elements, where p is a 4k + 1 prime. The adjacency relation and key properties follow from basic results in number theory, all of which will be reviewed with examples. Paley graphs are highly symmetric and have a variety of other interesting properties. In particular they are examples of concrete random graphs.

The main purpose of this talk is to discuss the improbable and not-yet-understood connection between the unique sum of squares representation p = x² + y² of these primes, and the structure of the corresponding Paley graph Pₚ.

This is joint work with Bobby Jacobs.

Let Qd be the hypercube of dimension d and let H and K be subsets of its vertex set V(Qd), called configurations in Qd. We say that K is an exact copy of H if there is an automorphism of Qd which sends H onto K. Let H be a configuration in Qd and let n ≥ d be an integer. We let λ(H, d, n) be the maximum, over all configurations A in Qn, of the fraction of sub-d-cubes R of Qn in which A ∩ R is an exact copy of H, and we define the d-cube density λ(H, d) of H to be the limit as n goes to infinity of λ(H, d, n).

We have determined λ(H, d) for 11 of the 14 configurations in Q3 (and have lower bound constructions close to flag algebra upper bounds for the others) and several of the 238 configurations in Q4, as well as for an infinite family of configurations. There are strong connections with the inducibility of graphs. We also have some recent results with Alon and Axenovich on determining λ(d, s), the limit as n goes to infinity of the maximum fraction, over all subsets A of the vertices of a large n-cube, of sub-d-cubes which have precisely s vertices in A.