Abstract:
The notion of Jacobian of a graph, also known as the Picard group, the critical group, the sandpile group
or the dollar group, was independently introduced by many authors ([1], [2],
[3], [4]). Given a graph one can define a Jacobian as the maximum Abelian group generated by flows
satisfying the first and the second Kirchhoff's laws. It is a crucial invariant of a finite graph. Its
order coincides with the number of spanning trees of the graph. It is also can be considered as a discrete
version of the Jacobian of a Riemann surface. The complete structure of the Jacobian is known only
for a few families of graphs. For instance, for the wheel graphs, the prism graphs, the Moebius ladders, the complete graphs and some others. The aim of this talk is to provide a general method to determine the structure of Jacobian
for an infinite family of circulant graphs.
The author is partially supported by the Laboratory of Quantum Topology of
Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020)
and RFBR grants 15-01-07906 and 16-31-50009.
References:
R. Cori, D. Rossin, On the sandpile group of dual graphs. European J. Combin. 21 (2000), no. 4, 447–459.
B. Baker, S. Norine, Harmonic morphisms and hyperelliptic graphs. Int. Math. Res. Notes 15 (2009), 2914–2955.
N. L. Biggs, Chip-firing and the critical group of a graph. J. Algebraic Combin. 9 (1999), no. 1, 25–45.
R. Bacher, P. de la Harpe, T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull. Soc. Math. France.
125 (1997), 167–198.
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