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In the article (Lovasz, section 1.3) mentions graph algebra structures on the set of formal linear combinations (over a field?) of a collection of graphs. He also mentioned quantum graphs as an example. My questions ( a question and one reference request ) are:

  1. What is the relationship between graph algebra and quantum graph?

  2. Could you suggest reference for graph algebra structures with the notion as in the linked article?

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Question 1. is, strictly speaking, ungrammatical, and vague in most interpretations. If by graph algebra you mean algebraic graph theory, then the question is hopelessly broad. If by question 1. you mean ``What is the relationship between graph algebras (in the sense of Lovász) and quantum graphs (also in the sense of Lovász), then the question is extremely specific and easy to answer: a quantum graph (in the sense of Lovász) is an element of a graph algebra (in the sense of Lovász). A definition of both these notions is given in e.g.

Hatami, H. and Norine, S.: Undecidability of linear inequalities in graph homomorphism densities. Journal of the American Mathematical Society, Vol. 24, No. 2 (2011), p.547--565

which would also be my answer to Question 2. This is a good article to start, rich in mathematical context (e.g. Artin's solution of Hilbert's seventeenth problem).

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  • $\begingroup$ Thank you for the reference, sorry for the badly worded question. $\endgroup$
    – mukhujje
    Commented Apr 21, 2017 at 5:58

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