Hankel matrices: From words to graphs (extended abstract)

Abstract

We survey recent work on the use of Hankel matrices H(f, □) for real-valued graph parameters f and a binary sum-like operation □ 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,□) 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.