#### Title

A new proof of an integral formula for counting perfect matchings in graphs

#### Presentation Type

Presentation

#### Faculty Mentor’s Full Name

P. Mark Kayll

#### Faculty Mentor’s Department

Mathematical Sciences

#### Abstract / Artist's Statement

With the modern proliferation of real-world networks, the almost quarter-millenium-old subject of graph theory has become increasingly important. A graph consists simply of a set of vertices and a set of edges, with every edge connecting two vertices. A perfect matching M in a graph G is a collection of pairwise disjoint edges of G with the property that each vertex of G is an end of an edge in M. The theory of matchings recently entered the limelight when the 2012 Nobel Prize in Economics was awarded to two pioneers of the subject (Alvin Roth and Lloyd Shapley). This talk will examine the question of how many perfect matchings a graph may contain. In [Combinatorica 1 (1981), 257--262], Godsil answered this question using an integral counting formula, and he proved it using an induction argument. Emerson and Kayll [Contributions to Discrete Mathematics 4 (2009), 89--93] mentioned the possibility of a different, more enlightening proof of Godsil's result, but they provided no details. Here I will present those details. The audience may appreciate the interplay between continuous and discrete mathematics.

A new proof of an integral formula for counting perfect matchings in graphs

UC 331

With the modern proliferation of real-world networks, the almost quarter-millenium-old subject of graph theory has become increasingly important. A graph consists simply of a set of vertices and a set of edges, with every edge connecting two vertices. A perfect matching M in a graph G is a collection of pairwise disjoint edges of G with the property that each vertex of G is an end of an edge in M. The theory of matchings recently entered the limelight when the 2012 Nobel Prize in Economics was awarded to two pioneers of the subject (Alvin Roth and Lloyd Shapley). This talk will examine the question of how many perfect matchings a graph may contain. In [Combinatorica 1 (1981), 257--262], Godsil answered this question using an integral counting formula, and he proved it using an induction argument. Emerson and Kayll [Contributions to Discrete Mathematics 4 (2009), 89--93] mentioned the possibility of a different, more enlightening proof of Godsil's result, but they provided no details. Here I will present those details. The audience may appreciate the interplay between continuous and discrete mathematics.