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Arithmetical structures on graphs. (English) Zbl 1428.05188
Summary: Arithmetical structures on a graph were introduced by D. J. Lorenzini in [Math. Ann. 285, No. 3, 481–501 (1989; Zbl 0662.14008)] as some intersection matrices that arise in the study of degenerating curves in algebraic geometry. In this article we study these arithmetical structures, in particular we are interested in the arithmetical structures on complete graphs, paths, and cycles. We begin by looking at the arithmetical structures on a multidigraph from the general perspective of M-matrices. As an application, we recover the result of Lorenzini about the finiteness of the number of arithmetical structures on a graph. We give a description on the arithmetical structures on the graph obtained by merging and splitting a vertex of a graph in terms of its arithmetical structures.
On the other hand, we give a description of the arithmetical structures on the clique-star transform of a graph, which generalizes the subdivision of a graph. As an application of this result we obtain an explicit description of all the arithmetical structures on the paths and cycles and we show that the number of the arithmetical structures on a path is a Catalan number.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C76 Graph operations (line graphs, products, etc.) 05E99 Algebraic combinatorics 11B65 Binomial coefficients; factorials; $$q$$-identities 11C20 Matrices, determinants in number theory 15B48 Positive matrices and their generalizations; cones of matrices
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