Algebraic Graph Theory Seminar - Chris Godsil

Monday, September 13, 2021 11:30 am - 11:30 am EDT (GMT -04:00)

Title: Tails and Chains

Speaker: Chris Godsil
Affiliation: University of Waterloo
Zoom: Contact Soffia Arnadottir


Physicists are interested in "graphs with tails"; these are constructed by choosing a graph X and a subset C of its vertices, then attaching a path of length n to each vertex in C. We ask what is the spectrum of such graph? What happen if n increases? We will see that the answer reduces to questions about the matrix

\[ M(\zeta) := (\zeta_\zeta^{-1})I - A -\zeta D \]

where D is the diagonal 01-matrix with D_{i,i}=1 if i is in C. (For physicists, the block of M(\zeta)^{-1} indexed by the entries of C determines the so-called scattering matrix of a quantum system, but we won't go there.)

A path is built by chaining copies of K_2 together. We consider what happen if we use some other graph in place of K_2.