Algebraic Graph Theory Seminar - Chris Godsil

Title: Tails and Chains

Abstract:

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.