Graph pebbling is a model aiming to solve the following resource allocation question: can a given supply meet some predetermined demand? A pebbling move is defined as taking two pebbles from a vertex and placing one pebble on a neighboring vertex. Given a graph G, the pebbling number π(G) is the minimum number of pebbles such that any configuration of that size can satisfy the target demand through a sequence of pebbling moves.

Graph pebbling on the hypercube was originally developed by Fan Chung as an alternate proof technique to resolve the Erdos-Lemke conjecture in number  ̋ theory. Since then, the pebbling number and variations have been studied for various classes of graphs. This talk will survey existing results and techniques used to determine the pebbling number, with a focus on the hypercube. It will also explore how generalizing techniques from the hypercube connects to Graham’s conjecture and further enriches the world of graph pebbling.

The reconstruction problem asks if we can uniquely identify a larger structure from its smaller substructures. I'll consider this problem in two contexts: rooted binary trees (a.k.a rooted phylogenetic tree shapes) and tanglegrams. For rooted binary trees, the smaller substructures are leaf-induced binary subtrees; we show such trees are reconstructable. A tanglegram consists of two rooted binary trees with the same number of leaves and a perfect matching between the leaves of the trees. We show that tanglegrams are reconstructable when at least one of the binary trees has that its internal vertices form a path that ends at the root.

This is a talk based on a research project done as a Sophomore at JMU, so it will assume no prior knowledge, with the hope of being accessible to anyone interested.

The critical group appears in many contexts, but the one I’ll focus on for this talk will be from the viewpoint of a fun game on a graph called ”chip-firing”. A configuration of a graph is an assignment of some number of chips to each vertex, and we can fire a vertex by sending 1 chip along each edge connected to it. We consider two configurations equivalent if we can get from one to the other through a series of chip-firing. The basic question is: Up to this equivalence, how many configurations can a graph G have?

In this talk, we will explore various examples of critical groups and briefly discuss their many connections to other fields of mathematics, including an appearance of an interesting poset, a beautiful analogue of the Riemann-Roch theorem, and a clever connection to rational points on an elliptic curve.

The distinguishing number of a graph is the least number of colors in a vertex coloring such that the only color-preserving automorphism is trivial. The determining number of a graph is size of a smallest set of vertices S such that the only automorphism fixing S (point-wise) is trivial. I will discuss results on these symmetry parameters for Mycielskian graphs and for various types of cube graphs. This is joint work with Debra Boutin, Sally Cockburn, Lauren Keough, Kat Perry, and Puck Rombach.

Let V be an n-dimensional vector space over a finite field of order q, and let L(V) be the partially ordered set (poset) of subspaces of V ordered by inclusion. The heuristic—going back to at least Gian-Carlo Rota—is that, as “q → 1”, the combinatorics of L(V) should be similar to that of 2 [n] , the poset of all subsets of a set with n elements—aka a Boolean lattice. In this talk, we will discuss two such instances, both regarding large families of subspaces that avoid certain configurations. What is the largest size of a family of kdimensional subspaces of V, that does not include three subspaces A, B, and C with A = (A∩B)⊕(A∩C)? What is the largest size of a family of subspaces (of any dimension) of V that does not include three subspaces A, B, and C with A ⊆ B ∩ C or B + C ⊆ A?

Normalized matching posets (aka LYM posets, the posets that satisfy the LYM inequality for antichains) are a class of posets that include subset and subspace lattices as well as divisor lattices. In this talk, we will discuss questions about partitioning a normalized matching poset into chains. Almost all such questions start with the poset of subsets of a finite set where many open questions remain. Given a finite (in our case normalized matching) poset P, and k positive integers that add up to |P|, can you find k chains (i.e., totally ordered subsets of the poset) that partition P and whose sizes are precisely the given positive integers? Particular cases of this question will be discussed.

At each step of the k-neighbor bootstrap percolation process on a graph, every uninfected vertex with at least k infected neighbors becomes infected. An initial set of infected vertices PERCOLATES if eventually every vertex becomes infected. Determining the minimum size of a percolating set under the 2-bootstrap model is a classic combinatorial puzzle -- the slower infection of the 3-bootstrap process seems harder to analyze.  Building on the work of Dukes, Noel, and Romer, we determine precisely the minimum size of a percolating in an (m x n) grid. We then consider the same question for toroidal grid graphs.

This is joint work with Alexander Clifton.