**UoP maths seminar (2013-14)**

**Some results on graphs and matrix spectra.**

*Speaker*: Murad Banaji

*Venue*: Wednesday 12th March 2014, 2pm, LG2.04a (refreshments provided)

*Abstract*: Algebraic graph theory involves deducing properties of graphs from algebraic objects - e.g. matrices, polynomials, groups - associated with the graphs. A result may go: "*if a graph G has spectral property Γ, then it belongs to class ***G**". Alternatively, we may be more interested in a linear algebraic question, but pass to graphs to help answer it. For example, we may deduce that all matrices associated with certain graphs have certain spectral properties. I'll present a couple of recent neat equivalences in this direction. The results take the form: "*a graph G belongs to class ***G** if and only if all matrices in class **M(**G**)** have propery M". This includes slightly unexpected characterisations of two well-studied classes of graphs, trees and caterpillars. The results are inspired by certain applications in dynamical systems.