Sharp orders of arithmetical critical groups via canonical lattice simplices
Henry Zweiman
September 2026
Abstract
We identify the critical group of an arithmetical structure on a connected simple graph with the vertex-lattice quotient of a canonical lattice simplex. Applying the sharp multiplicity theorem of Averkov, Kasprzyk, Lehmann and Nill, we determine the largest possible critical-group order among all such structures on graphs with a fixed number of vertices. Complete graphs attain the bound. The same bound holds for stars with a fixed number of leaves, proving the largest-order conjecture of Archer, Diaz-Lopez, Glass and Louwsma. We also determine the maximizing arithmetical structures on complete graphs and stars.
1. Introduction and main result
Let be a connected simple graph on , with , and let be its adjacency matrix. An arithmetical structure on is a pair of positive integer vectors such that is primitive and
Its critical group is the finite abelian group
The ordinary graph Laplacian is the case . Arithmetical structures allow other diagonal entries, producing critical groups whose orders can be much larger than the numbers of spanning trees of the underlying graphs.
Write and for Sylvester’s sequence, and define
For stars we use the convention that has leaves and hence vertices.
Theorem 1.
For every arithmetical structure on a connected simple graph with vertices,
For each , the bound is attained by an arithmetical structure on the complete graph . Moreover, is the largest critical-group order among arithmetical structures on the star .
Archer, Diaz-Lopez, Glass and Louwsma [1] determine critical groups of arithmetical structures on stars and complete graphs. Their Conjecture 24 asserts the last conclusion of Theorem 1 for , together with its complete-graph counterpart. Their notation is and , so that . Thus Theorem 1 proves that conjecture and gives a sharp bound over all connected simple graphs with a fixed number of vertices.
The geometric input is the sharp canonical-simplex multiplicity bound of Averkov, Kasprzyk, Lehmann and Nill [2]. We use their theorem, including its equality classification, as an external result; we do not give a new proof of that bound. Taking the convex hull of Laplacian columns is the earlier construction of Braun and Meyer [4]. Balletti, Hibi, Meyer and Tsuchiya [3] show that ordinary Laplacian simplices of simple strongly connected digraphs have a unique interior lattice point. Our observation is that a symmetric arithmetical Laplacian gives such a simplex in the saturated lattice perpendicular to its positive kernel vector, and that its critical group is exactly the vertex-lattice quotient. This makes the multiplicity theorem applicable to the graph-order problem.
2. The simplex and its vertex-lattice quotient
For a rank- lattice , a full-dimensional lattice simplex is called canonical if its unique interior lattice point is the origin. Its multiplicity is the index
In particular, the ambient lattice is part of this definition.
Lemma 2.
Let be an arithmetical structure on a connected simple graph with vertices. Let be the columns of and set
Then is a canonical -dimensional simplex in , and
Proof.
Put . The matrix is the weighted Laplacian with edge weights , because
Consequently, for every ,
Connectedness implies that has rank and . Symmetry puts each in . The only linear relation among the columns is a multiple of . Since , no nonzero relation is an affine relation, so the columns are affinely independent. The positive coefficients express as an interior point of their simplex.
Every coordinate of every is at least . For each , column has positive th coordinate : every vertex has a neighbor, and is the positive sum of the neighboring -entries. An interior point of uses all vertices with strictly positive coefficients. Each of its coordinates is therefore strictly greater than . An interior lattice point has all ; the equation and positivity of force . This proves that is canonical. Its vertices are primitive, since each column has an entry at a neighboring vertex.
Finally, primitivity of gives an exact sequence
The left term is finite because . It equals the torsion subgroup of the middle term: torsion maps to zero in , and every element of the finite kernel is torsion. The columns generate , giving the claimed isomorphism.
We state the precise geometric input in lattice language.
Theorem 3 (Averkov–Kasprzyk–Lehmann–Nill).
Let be a canonical -dimensional lattice simplex, where . Then
Up to a lattice isomorphism, the simplex attaining equality is unique. It is
where in the last line and the displayed models have ambient lattice .
This is Proposition 2.8 and Theorem 2.9 of [2]. Combining it with Lemma 2, with , proves the upper bound in Theorem 1.
3. Attainment and the star bound
We record an elementary order formula. If is a symmetric positive semidefinite integral matrix of rank with primitive null vector , then
Indeed the rank-one adjugate is a rational multiple of . All its entries are integral, and , so the multiple is an integer. The gcd of the -minors is the order of the torsion cokernel by Smith normal form. Positivity of the nonzero eigenvalues fixes the positive sign in (2).
Let be positive integers with
Put , and . All , and is primitive: for each prime dividing , some denominator has its full valuation. Since , these data define an arithmetical structure on with
where is the all-ones matrix. Deleting row and column and applying the determinant lemma gives the principal cofactor
This is also the order consequence of the group formula in [1].
For , take for every ; formula (4) gives . For , take
giving order . For , put
The telescoping identity proves (3). Here , and (4) gives
These are the extremal constructions already proposed in [1]; their optimality follows from the upper bound.
Now consider an arithmetical structure on . Denote its center by , put at its leaves, and let be its central label. The arithmetical equations give and
Primitivity implies . The principal cofactor obtained by deleting the center is , so (2) gives
Replace each by . The new tuple satisfies (3), has least common multiple , and determines a complete-graph structure of order
The complete-graph bound therefore proves the stronger inequality
This scaling is the order calculation underlying [1, Proposition 20]. Conversely, each maximizing unit-fraction tuple above defines a star structure with and order by (6). This completes the proof of Theorem 1.
4. Maximizing structures on complete graphs and stars
The equality classification in the geometric theorem gives more than the numerical answer to the conjecture.
Corollary 4.
Up to permutation of vertices, the maximizing arithmetical structure on is unique for each . It is given by and , where and
with . For with , the maximizing structure is likewise unique up to permutation of leaves: its central label is , its leaf labels are the displayed , and .
Proof.
For a complete-graph structure put and . The equations give , hence . The barycentric coordinates of in the simplex of Lemma 2 are . Lattice isomorphisms preserve these coordinates up to permutation. The equality models in Theorem 3 have reciprocal barycentric coordinates precisely the tuples in the statement. For the last model, the nonzero-axis vertices have coordinates , and the remaining coordinate is by the telescoping identity. The low-dimensional models give the other two tuples. Thus equality determines up to permutation. It then determines and the primitive vector uniquely.
For a star with , equality in (7) forces . Its leaf tuple therefore determines a complete-graph structure of the same order, so the first part applies. The displayed constructions establish existence in every case.
For every arithmetical critical group is trivial, and uniqueness is not asserted. Nor do we classify all graphs attaining the universal bound in Theorem 1. The numerical bound and its attainment need no such classification.
References
[1] K. Archer, A. Diaz-Lopez, D. Glass, and J. Louwsma, Critical groups of arithmetical structures on star graphs and complete graphs, Electron. J. Combin. 31 (2024), no. 1, Paper P1.5. doi:10.37236/11729.
[2] G. Averkov, A. Kasprzyk, M. Lehmann, and B. Nill, Sharp bounds on fake weighted projective spaces with canonical singularities, in Varieties, Polyhedra, Computation, EMS Series of Congress Reports, EMS Press, 2025, pp. 319–334. doi:10.4171/ecr/22/10. Preprint: arXiv:2105.09635.
[3] G. Balletti, T. Hibi, M. Meyer, and A. Tsuchiya, Laplacian simplices associated to digraphs, Ark. Mat. 56 (2018), 243–264. doi:10.4310/ARKIV.2018.v56.n2.a3.