Developments on Spectral Characterizations of Graphs

In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs

A Closed-Form Shave from Occam's Quantum Razor: Exact Results for Quantum Compression

The causal structure of a stochastic process can be more efficiently transmitted via a quantum channel than a classical one, an advantage that increases with codeword length. While previously difficult to compute, we express the quantum advantage in closed form using spectral decomposition, leading to direct computation of the quantum communication cost at all encoding lengths, including infinite. This makes clear how finite-codeword compression is controlled by the classical process' cryptic order and allows us to analyze structure within the length-asymptotic regime of infinite-cryptic order (and infinite Markov order) processes.Comment: 21 pages, 13 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/eqc.ht