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Let G be a finite, connected graph. An arithmetical structure on G is an assignment of positive integers to the vertices such that, at each vertex, its label divides the sum of the labels at adjacent vertices (counted w/ multiplicity). Arithmetical structures naturally generalize the notion of the Laplacian matrix of a graph. They also appear in the context of arithmetical geometry.
In this talk, I will present a general introduction to the topic, we will focus on the combinatorics of arithmetical structures on certain graph families including some counting results that relate arithmetical structures on a path graph and Catalan numbers. We will also discuss some results about the associated finite Abelian group of an arithmetical structure.
What is an arithmetical structure? Sponsored by the Meyer Fund
What is an arithmetical structure?