## A Combinatorial Description for the Coefficients of the Adjacency Characteristic Polynomial of a Hypergraph

- Feb. 2, 2018
- 2:30 p.m.
- LeConte 312

## Abstract

As the title suggests, we prove a formula for the coefficients of the normalized adjacency matrix of a k-uniform hypergraph. This formula is a generalization of the graph case which computes the codegree *d* coefficient of the adjacency characteristic polynomial as a function of the closed walks on *d* edges within the graph. For the case of hypergraphs, we call the enumerated hypergraphs Veblen hypergraphs and prove that each occurrence of a Veblen hypergraph contributes a fixed amount to a particular coefficient. We define this amount to be the associated coefficient of the hypergraph. The purpose of this talk is threefold: prove the aforementioned formula, provide a closed formula for the associated coefficient of the simplex, and compute the associated coefficient of all small 3-uniform Veblen hypergraphs.

Joint work with Joshua Cooper of the University of South Carolina.