Algebraic Graph Theory Seminar - Mahsa N. Shirazi

Title: On the eigenvalues of the perfect matching derangement graph

Abstract:

The perfect matching derangement graph $M_{2n}$ is defined to be the graph whose vertices are all the perfect matchings of the complete graph $K_{2n}$, and two vertices are adjacent if they contain no common edges. The graph $M_{2n}$ is part of a larger study on the analogs of the Erdős-Ko-Rado theorem, and recently there have been interesting works on $M_{2n}$ and its eigenvalues.

In 2016, Godsil and Meagher conjectured that the graph $M_{2n}$ possesses the alternating sign property. Recently, it was shown by Koh, Ku, and Wong that this conjecture is true, and they presented a recurrence formula for the eigenvalues of $M_{2n}$.

The eigenvalues of the graph $M_{2n}$ are indexed by even integer partitions. In our work, we calculate the eigenvalues of $M_{2n}$ corresponding to some classes of partitions: [2k, 2, 2, …,2 ], [4, 4, …, 4, 2, 2, …,2], and [2k, 2n-2k]. We also propose a conjecture on the order of the absolute values of the eigenvalues corresponding to the aforementioned classes of partitions.

This is joint work with Karen Meagher.