TY - JOUR
T1 - Sandpile groups of Cayley graphs of Fr2
AU - Gao, Jiyang
AU - Marx-Kuo, Jared
AU - McDonald, Vaughan
AU - Yuen, Chi Ho
N1 - Publisher Copyright:
© 2024 The Author(s). Published with license by Taylor & Francis Group, LLC.
PY - 2024
Y1 - 2024
N2 - The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of (Formula presented.), focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for “generic” Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n–1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries.
AB - The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of (Formula presented.), focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for “generic” Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n–1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries.
KW - Cayley graph
KW - sandpile group
UR - http://www.scopus.com/inward/record.url?scp=85193235775&partnerID=8YFLogxK
U2 - 10.1080/00927872.2024.2347582
DO - 10.1080/00927872.2024.2347582
M3 - Article
AN - SCOPUS:85193235775
SN - 0092-7872
VL - 52
SP - 4459
EP - 4479
JO - Communications in Algebra
JF - Communications in Algebra
IS - 10
ER -