Of
Graph Theory and
Combinatorics
2004-05 SCHEDULE
Tues, Nov 9 - Colin Starr (
“Unipancyclic
Matroids”
Abstract: A unipancyclic (UPC) graph is a simple graph having exactly one cycle of each size from 3 to n,
where n is the number of vertices of the graph. Klas Markstrom proved by a computer search that there are
only a few UPC graphs with 56 or fewer vertices; a graduate student at Louisiana Tech is working on expanding
this result. Dr. Galen Turner and I have asked the same question of matroids and found a small UPC matroid
that is not graphic. I will discuss our techniques and share some MAPLE code I wrote to aid with this problem.
Mon, Dec 6 - Chuck Dunn (
"Graph
Coloring Games"
Abstract: Let G be a graph, let X be a set of r colors, and let d be a nonnegative integer. Two players, Alice
and Bob, alternate coloring the vertices of G using legals colors from X. A color x is legal for an uncolored vertex u if,
after u is colored,
the subgraph induced by all vertices colored x has maximum degree at most
d.
if all vertices of G
are eventually colored. Otherwise, Bob wins. The least r such that
called the d-relaxed game chromatic number of G. We will discuss this game and the known bounds on the associated
parameters with trees, outerplanar graphs, and partial k-trees. We will also consider other variations of the game.
Mon, Feb 28 - John Caughman (
“Laplacians
of Directed Graphs”
Abstract: Let G denote a directed graph with adjacency matrix Q and in-degree matrix D. We
consider the Kirchhoff matrix L = D - Q, sometimes referred to as the directed Laplacian. A classical
result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals
the number of connected components of G. We generalize this result to directed graphs G, showing
first that the algebraic and geometric multiplicities of 0 are equal, and then deriving a natural
basis for the corresponding eigenspace. A graph-theoretic property determines the dimension of
this eigenspace, namely, the number of reaches of the directed graph G. This result is stated and
proved in the more general context of stochastic matrices, and applies equally well to directed
graphs with non-negative edge weights.
Mon, April
“The
Mathematics of Google”
7:00pm,
Room 348, Neuberger Hall
Abstract: The main reason for Google's popularity is its uncanny ability to quickly provide particularly useful search
results through the use of their PageRank algorithm. Although there have been many attempts to manipulate these
search results, it is safe to assume that most people are not aware that this algorithm lies at the intersection of the fields of
graph theory, linear algebra, and probability theory. We will motivate the development of the PageRank algorithm by
tracing the original paper by the developers Larry Page and Sergey Brin and ultimately learn that the heart of the
algorithm involves efficiently finding eigenvectors for dominant eigenvalues for stochastic matrices associated
with the Web-graph.
Mon, May 9 – Jeremy LeCrone (Pacific
University)
“A Matroid on a Finite Group: Proof and Consequences”
Abstract: I will be presenting results that I have generated as
part of my senior project at
Using a convenient partition of the elements of a general group, I have developed a definition of a matroidon a finite group.
A proof of this result will be presented. Then I willintroduce a few commonly encountered matroid quantities and
demonstrate how they apply to some familiar finite groups. A second, possible, definition for a matroid on a group
will be explored and the counterexample presented. Finally, a graphic representation of the matroid will be introduced
and the consequences of taking the dual of this graph on the structure of the matroid explored.
Back to my home page.