Homocyclic Jacobians of planar biconnected graphs

Nifa, Anass

doi:10.5281/zenodo.22859833

Preprint · v1

Homocyclic Jacobians of planar biconnected graphs

Anass Nifa5 May 2026

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Abstract

For every $r\ge1$ , we construct a simple planar biconnected graph with $2r+2$ vertices, $4r+1$ edges, maximum degree at most four, and Jacobian $(\mathbb{Z}/8\mathbb{Z})^r$ . This disproves the bounded-multiplicity conjecture of Gaudet, Jensen, Ranganathan, Wawrykow and Weisman. We give explicit generators and compute the monodromy pairing: its matrix is $B_r/8$ modulo $\mathbb{Z}$ , where $B_r$ is tridiagonal with diagonal $4,\ldots,4,5$ and adjacent entries $-1$ . For every finite tree $T$ with at least two vertices and every integer $m\ge2$ , we also prove $\operatorname{exp}\operatorname{Jac}(T[\overline{K_m}]) =m^2\operatorname{lcm}_{v\in V(T)}\deg_T(v).$ For paths with at least three vertices, these products have vertex connectivity $m$ and exponent $2m^2$ . They disprove the bounded-exponent finiteness conjecture attributed to Baker and Shokrieh, including its analogue for connected regular matroids.

MSC 2020

05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
05C05: Trees
05C40: Connectivity
05B35: Combinatorial aspects of matroids and geometric lattices

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22859833
All versions: 10.5281/zenodo.22859832
Subjects: Graph Jacobian; critical group; homocyclic group; monodromy pairing; exponent; vertex connectivity
Language: English

Cite this paper

Anass Nifa. Homocyclic Jacobians of planar biconnected graphs. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22859833

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References

As listed in the manuscript bibliography.

[1]
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Source · arXiv v1 · arXiv v2
[9]
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[reference-1] [1]
R. Bacher, P. de la Harpe and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France 125 (1997), no. 2, 167–198. 10.24033/bsmf.2303.
DOI

[reference-2] [2]
M. Baker and F. Shokrieh, Chip-firing games, potential theory on graphs, and spanning trees, J. Combin. Theory Ser. A 120 (2013), no. 1, 164–182. 10.1016/j.jcta.2012.07.011.
DOI

[reference-3] [3]
N. Biggs, Algebraic potential theory on graphs, Bull. London Math. Soc. 29 (1997), no. 6, 641–682. 10.1112/S0024609397003305.
DOI

[reference-4] [4]
X. Dong, W. Guo and G. Jiang, On the critical group of the k-partite graph, preprint (2024), arXiv:2409.02654v1.
arXiv v1

[reference-5] [5]
L. Gaudet, D. Jensen, D. Ranganathan, N. Wawrykow and T. Weisman, Realization of groups with pairing as Jacobians of finite graphs, Ann. Comb. 22 (2018), no. 4, 781–801. 10.1007/s00026-018-0406-0. arXiv:1410.5144v1 (2014), Conjecture 34; v2 (2017), Conjecture 36.
DOI · arXiv v1 · arXiv v2

[reference-6] [6]
W. W. Hager, Updating the inverse of a matrix, SIAM Rev. 31 (1989), no. 2, 221–239. 10.1137/1031049.
DOI

[reference-7] [7]
B. Jacobson, A. Niedermaier and V. Reiner, Critical groups for complete multipartite graphs and Cartesian products of complete graphs, J. Graph Theory 44 (2003), no. 3, 231–250. 10.1002/jgt.10139.
DOI

[reference-8] [8]
H. Lheem, D. Li, C. J. Quines and J. Zhang, Exponents of Jacobians of graphs and regular matroids, Rose-Hulman Undergrad. Math. J. 21 (2020), no. 2, Article 6. arXiv:1910.06442v1 (2019); v2 (2021), whose conjecture numbering is used here.
Source · arXiv v1 · arXiv v2

[reference-9] [9]
D. Lorenzini, Smith normal form and Laplacians, J. Combin. Theory Ser. B 98 (2008), no. 6, 1271–1300. 10.1016/j.jctb.2008.02.002.
DOI

[reference-10] [10]
M. C. Marino and N. Zagaglia Salvi, Generalizing double graphs, Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 85 (2007), no. 2, Article C1A0702002. 10.1478/C1A0702002.
DOI

[reference-11] [11]
E. Munarini, C. Perelli Cippo, A. Scagliola and N. Zagaglia Salvi, Double graphs, Discrete Math. 308 (2008), nos. 2–3, 242–254. 10.1016/j.disc.2006.11.038.
DOI

[reference-12] [12]
S. Pirzada and H. A. Ganie, Energy, Laplacian energy of double graphs and new families of equienergetic graphs, preprint (2013), arXiv:1310.3204v1.
arXiv v1

[reference-13] [13]
F. Shokrieh, The monodromy pairing and discrete logarithm on the Jacobian of finite graphs, J. Math. Cryptol. 4 (2010), no. 1, 43–56. 10.1515/JMC.2010.002.
DOI