Gonality of Circulant Graphs

• Circulant graphs are vertex-transitive structures defined by cyclic adjacency, with gonality measuring the minimal chip configuration achieving nonnegative balance.
• The study employs chip-firing dynamics and symmetric divisor constructions to establish universal gonality bounds that hold regardless of the number of vertices.
• Computational techniques exploiting graph symmetry yield sharp gonality values for families like Harary graphs and reveal connections to tropical geometry.

Circulant graphs are a class of vertex-transitive graphs defined by cyclic adjacency relations, specified by their number of vertices and a fixed adjacency list. Their symmetry makes them a compelling subject for the paper of divisorial gonality, an invariant directly connected to chip-firing dynamics and with analogies to the classical theory of linear series on algebraic curves. The following presents a rigorous and comprehensive account of the gonality of circulant graphs, spanning foundational definitions, theoretical frameworks, universal bounds, computational approaches, and key implications in modern research.

1. Gonality: Definition and Framings

Gonality in the context of graphs is a combinatorial analogue of the gonality of algebraic curves. For a finite undirected graph $G$, a divisor is an element of $\mathbf{Z}^{V(G)}$, interpreted as an integer assignment (chips) to each vertex. The divisorial gonality $\operatorname{gon}(G)$ is the smallest degree $d$ for which some effective divisor $D$ of degree $d$ has rank at least one. In formal chip-firing language, this means for every local introduction of "debt," there is a sequence of allowable chip-firing moves (including subset-firing) to eliminate all debt while maintaining nonnegative chips throughout.

For circulant graphs $\mathrm{Ci}_n(J)$ determined by vertex count $n$ and adjacency list $J = \{j_1,\ldots,j_k\}$ (with vertices $v_1,\ldots,v_n$, edges $v_\alpha \to v_{(\alpha \pm j_i)\bmod n}$), the gonality is central to both combinatorics and tropical geometry.

2. Chip-Firing Moves and Divisor Dynamics

Chip-firing on a graph $G$ is defined by the Laplacian matrix $Q = D(G) - A(G)$, where firing a subset $S \subseteq V(G)$ changes a divisor $D$ to $D' = D - Q 1_S$. The game is won if, for a configuration $D$ of sufficient degree, all possibilities of placing a single unit of debt at an arbitrary vertex can eventually be corrected by legal firing moves.

In circulant graphs, the combinatorial regularity enables the construction of symmetric divisors and prescribed chip movement. The explicit formula for the constructed divisor $D$ ensuring positive rank in $\mathrm{Ci}_n(J)$ is

$$D(v_\alpha) = \begin{cases} \sum_{i=1}^k \max\{ 0, j_i - |j_k - \alpha | \} & \text{if } \alpha \in [1,2j_k-1], \ D(v_{n-\alpha+1}) & \text{if } \alpha \in [n-2j_k+1,n], \ 0 & \text{otherwise}. \end{cases}$$

This placement guarantees, via subset-firing moves, that every vertex eventually receives a chip, exploiting the circulant's cyclic symmetry (Cenek et al., 7 Aug 2025).

3. Universal Upper Bound and Explicit Gonality Values

A principal result establishes a universal upper bound for the gonality of any connected circulant graph with a fixed adjacency list: $$\operatorname{gon}(\operatorname{Ci}_n(J)) \leq 2 \sum_{i=1}^{k} j_i^2,$$ where $k$ is the length of the adjacency list. This uniform upper bound is independent of $n$, tightly governing gonality for infinite families as $n$ varies. The proof is constructive and uses explicit chip-firing strategies.

For Harary graphs $H_{k,n}$ (degree-$k$ circulant graphs), the paper determines sharp gonality values—especially for the $4$-regular case: $$\operatorname{gon}(H_{4,n}) = \begin{cases} \left\lfloor \frac{n}{4} + \left\lfloor \frac{n+1}{4} \right\rfloor \right\rfloor & \text{if } n < 16, \ 10 & \text{if } n \geq 16. \ \end{cases}$$ For large enough $n$, the gonality stabilizes regardless of further growth in the graph, a phenomenon also holding for antiprism graphs as a special case (Cenek et al., 7 Aug 2025).

4. Moduli of Gonality, Parameter Counting, and Metric Graphs

In the broader theory, gonality is characterized in metric graphs by tropical morphisms to trees subject to Riemann–Hurwitz conditions: $k - 2 \geq m_\varphi(v) (l - 2),$ for local valency $k$ and local map degree $m_\varphi(v)$ to tree vertices of valency $l$. The locus of graphs of gonality at most $d$ in the moduli space of genus-$g$ graphs has dimension $\min \{ 3g-3, 2g + 2d - 5 \}$ (Cools et al., 2016).

For circulants, the high automorphism group can both constrain and simplify such gluing constructions, though one must adapt generic parameter-count arguments to the cyclic structure.

5. Connections to Divisor Theory, Weak Equivalence, and Degree Polynomials

Degree distribution polynomials classify circulant graphs up to weak equivalence and encode the frequency of graphs with given regularity: $Y_{A}(x) = \sum_k a_k(A) x^k,$ where $a_k(A)$ counts weak equivalence classes with vertex degree $k$. These polynomials inform the combinatorial diversity within the family and indirectly constrain gonality, as regular graphs typically admit only higher-degree divisors of positive rank, raising potential lower bounds (Kim et al., 2014). Gonality is invariant under isomorphism within weak equivalence classes, justifying this classification.

6. Specialized Graph Families, Construction Methods, and Gonality Sequences

Gonality three is characterized via the existence of nondegenerate harmonic morphisms to trees of degree three or via cyclic automorphisms of order three with tree quotients, in highly connected graphs satisfying the "zero-three condition." For circulant graphs—often 3-vertex and 3-edge connected—the classification and explicit construction via triplication of trees is applicable (Aidun et al., 2018).

The gonality sequence

$\gamma_r(G) = \min \{ \deg(D) : r(D) \ge r \}$

for $r\ge 1$ is determined uniquely by genus and the first gonality for $g\leq 5$; explicit constructions realize any pair $(\gamma_1,\gamma_2)$ in the feasible range (Aidun et al., 2020). The burning algorithm and its modifications provide practical computational tools for gonality detection in symmetric graph families, including circulants.

7. Computational Approaches and Implications

Chip-firing dynamics remain central to computationally determining gonality. Specialized interfaces and exploitation of circulant symmetry vastly reduce the complexity of these computations (Cenek et al., 7 Aug 2025). The NP-hardness of gonality and multiplicity-free gonality remains a challenge (Dean et al., 2021), although for many symmetric circulants gonality equals vertex-connectivity and multiplicity-free gonality, a fact leveraged for practical bounds.

Gorenstein properties in circulant graphs are tightly linked to explicit numerical relations (such as $n=2d+3$ for $C_n(1,...,d)$) and often coincide with highly symmetric structures, suggesting regularity constraints correspond to low gonality (Nikseresht et al., 2019).

The results generalize beyond specific circulants: the universal upper bound for gonality holds for all families with fixed adjacency lists, and computational techniques extend to broader classes via symmetry reduction.

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Summary Table: Universal Gonality Bounds for Circulant Graphs

Family	Gonality Bound	Stabilization Threshold
General $\mathrm{Ci}_n(J)$	$2 \sum_{i=1}^{k} j_i^2$	Independent of $n$
$H_{4,n}$ (Harary)	$\leq 10$ (Exacts: see formula)	$n \geq 16$
Antiprism, $n$ large	$10$	$n \geq 16$

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Conclusion: Gonality in circulant graphs is governed by chip-firing techniques, symmetric divisor constructions, universal bounds rooted in adjacency data, and rigorous computational methods. These results elucidate the interaction of combinatorial structure and arithmetic invariants, establishing both broad general theories and sharp values for important subclasses. The landscape remains fertile for further investigation, especially into explicit divisor realizations, deeper connections with algebraic invariants, and algorithmic refinement.