Abstract
We survey recent work on the use of Hankel matrices \(H(f, \Box )\) for real-valued graph parameters \(f\) and a binary sum-like operation \(\Box \) 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 \(\mathrm {MSOL}\) and show how \(\mathrm {MSOL}\)-definability can be replaced by the assumption that \(H(f, \Box )\) has finite rank. In contrast to \(\mathrm {MSOL}\)-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.
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