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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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References:
[1] Benjamin Braun, Scott Corry, Art Duval, Glass Darren, Lionel Levine, Jeremy Martin, Gregg Musiker, private communication.
[2] Biggs, Norman, Algebraic potential theory on graphs, Bull. Lond. Math. Soc., 29, 6, 641-682, (1997), MR 1468054 · Zbl 0892.05033
[3] Berman, Abraham; Plemmons, Robert J., Nonnegative matrices in the mathematical sciences, Classics in Applied Mathematics, vol. 9, (1994), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA, revised reprint of the 1979 original, MR 1298430 · Zbl 0815.15016
[4] Corrales, Hugo; Valencia, Carlos E., On the critical ideals of graphs, Linear Algebra Appl., 439, 12, 3870-3892, (2013), MR 3133463 · Zbl 1295.13028
[5] Corrales, Hugo; Valencia, Carlos E., Arithmetical structures on graphs with connectivity one, J. Algebra Appl., 17, 8, (2018) · Zbl 1428.05188
[6] Dickson, Leonard Eugene, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, Amer. J. Math., 35, 4, 413-422, (1913), MR 1506194 · JFM 44.0220.02
[7] Guzmán, Johnny; Klivans, Caroline, Chip-firing and energy minimization on M-matrices, J. Combin. Theory Ser. A, 132, 14-31, (2015), MR 3311336 · Zbl 1307.05032
[8] Godsil, Chris; Royle, Gordon, Algebraic graph theory, Graduate Texts in Mathematics, vol. 207, (2001), Springer-Verlag New York, MR 1829620 · Zbl 0968.05002
[9] Lorenzini, Dino J., Arithmetical graphs, Math. Ann., 285, 3, 481-501, (1989), MR 1019714 · Zbl 0662.14008
[10] Lorenzini, Dino J., Groups of components of Néron models of Jacobians, Compos. Math., 73, 2, 145-160, (1990), MR 1046735 · Zbl 0737.14008
[11] Lorenzini, Dino, Smith normal form and Laplacians, J. Combin. Theory Ser. B, 98, 6, 1271-1300, (2008), MR 2462319 · Zbl 1175.05088
[12] O’Carroll, Liam; Planas-Vilanova, Francesc; Villarreal, Rafael H., Degree and algebraic properties of lattice and matrix ideals, SIAM J. Discrete Math., 28, 1, 394-427, (2014), MR 3180844 · Zbl 1334.13017
[13] Stanley, Richard P., Catalan numbers, (2015), Cambridge University Press New York, MR 3467982 · Zbl 1317.05010
[14] Varga, Richard S., Geršgorin and his circles, Springer Series in Computational Mathematics, vol. 36, (2004), Springer-Verlag Berlin, MR 2093409 · Zbl 1057.15023
[15] Carlos E. Valencia, Ralihe R. Villagrán, Arithmetical structures on graphs with twins, manuscript.
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