IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

• Feb. 1, 2019
• 2:30 p.m.
• LeConte 312

## Abstract

The task of determining a combinatorial description for the first four coefficients of the adjacency characteristic polynomial of a graph is traditionally given as a homework problem. In this talk, we provide a solution to the inciting problem (via the Harary-Sachs Theorem) and present a generalization of the theorem for hypergraphs. In doing so, we are able to examine the first $k$+1 coefficients of the characteristic polynomial of a $k$-uniform hypergraph. It was shown by Cooper and Dutle that the codegree-($k$+1) coefficient can be expressed as a certain constant, which depends on $k$, multiplied by the number of simplexes in the hypergraph. We prove this and provide computational insight into the aforementioned constant. We conclude by presenting a general formula for the first seven coefficients of a 3-uniform hypergraph. This is joint work with Joshua Cooper.

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