Hankel Matrices: From Words to Graphs

We survey recent work on the use of Hankel matrices H(f,2) for real-valued graph parameters f and a binary sum-like operation 2 on labeled graphs such as the disjoint union and various gluing operations of pairs of laeled graphs. Special cases deal with real-valued word functions. We start with graph parameters definable in Monadic Second Order Logic MSOL and show how MSOL-definability can be replaced by the assumption that H(f,2) has finite rank. In contrast to MSOL-definable graph parameters, there are uncountably many graph parameters f with Hankel matrices of finite rank. We also discuss how real-valued graph parameters can be replaced by graph parameters with values in commutative semirings. In this talk we survey recent work done together with the first author’s former and current graduate students B. Godlin, E. Katz, T. Kotek, E.V. Ravve, and the second author on the definability of word functions and graph parameters and their Hankel matrix. There are three pervasive themes. – Definability of word functions and graph parameters f in some logical formalism L which is a fragment of Second Order Logic SOL, preferably Monadic Second Order Logic MSOL, or CMSOL, i.e., MSOL possibly augmented with modular counting quantifiers; – Replacing the definability of f by the assumption that certain Hankel matrices have finite rank; and – Replacing the field of real numbers R by arbitrary commutative rings or semirings S.