Seminar series
Date
Mon, 13 Oct 2025
16:00
16:00
Location
C3
Speaker
Cedric Pilatte
Organisation
Mathematical Insitute, Oxford
Understanding the eigenvalues of the adjacency matrix of a (possibly weighted) graph is a problem arising in various fields of mathematics. Since a direct computation of the spectrum is often too difficult, a common strategy is to instead study the trace of a high power of the matrix, which corresponds to a high moment of the eigenvalues. The utility of this method comes from its combinatorial interpretation: the trace counts the weighted, closed walks of a given length within the graph.
However, a common obstacle arises when these walk-counts are dominated by trivial "backtracking" walks—walks that travel along an edge and immediately return. Such paths can mask the more meaningful structural properties of the graph, yielding only trivial bounds.
This talk will introduce a powerful tool for resolving this issue: the non-backtracking matrix. We will explore the fundamental relationship between its spectrum and that of the original matrix. This technique has been successfully applied in computer science and random graph theory, and it is a key ingredient in upcoming work on the 2-point logarithmic Chowla conjecture.