In the paper

for each (finite simple) graph $\Gamma$, bivariate rational generating functions $W_\Gamma^\pm(q,t) \in \mathbf Q(q,t) \cap \mathbf Q(q)[\![t]\!]$ are defined. These generating functions encode average sizes of kernels in modules of (anti-)symmetric matrices with support constraints defined by $\Gamma$. The package Zeta for SageMath includes an implementation of an algorithm for explicitly computing $W_\Gamma(q,t)$. The following lists of the rational functions $W_\Gamma^\pm(q,t)$ for all graphs with at most $7$ vertices were computed using Zeta. In the lists, an “ask[±] zeta function” refers to an instance of $W_\Gamma^\pm(q,t)$. These functions are provided as SageMath input and, for graphs with at most six vertices, also rendered using MathJax.

Description of graphs | SageMath input | Rendered using MathJax |
---|---|---|

At most 5 vertices | W_1-5_inline.html | W_1-5_inline_latex.html |

At most 5 vertices (connected graphs only) | W_1-5_connected_inline.html | W_1-5_connected_inline_latex.html |

6 vertices | W_6_inline.html | W_6_inline_latex.html |

6 vertices (connected graphs only) | W_6_connected_inline.html | W_6_connected_inline_latex.html |

7 vertices | W_7.html | |

7 vertices (connected graphs only) | W_7_connected.html |

In the files above, graphs are specified using the graph6 format. Plots were created using SageMath.

Author: Tobias Rossmann. Last modified: 15 November 2021