Abelian Ramanujan Graphs

• Abelian Ramanujan graphs are highly regular Cayley graphs of finite abelian groups whose nontrivial eigenvalues are bounded by 2√(k-1), ensuring optimal spectral properties.
• They are constructed using symmetric generating sets in abelian groups and rings, with notable examples including Gaussian integer and norm-one graphs that yield infinite families.
• Their eigenvalues are computed via character sums and bounds like Deligne’s, which not only provide spectral optimality but also facilitate applications in coding theory and additive combinatorics.

An Abelian Ramanujan graph is a highly regular and optimal expander graph arising as a Cayley graph of a finite abelian group, whose nontrivial adjacency eigenvalues are bounded in absolute value by the Ramanujan bound $2\sqrt{k-1}$, where $k$ denotes the degree of regularity. Such graphs are distinguished by their algebraic construction — typically via symmetric generating sets in abelian groups or rings — and possess both spectral optimality and significant combinatorial structure, making them central objects in additive combinatorics, coding theory, and spectral graph theory.

1. Fundamental Definitions and Canonical Constructions

A Cayley graph $\mathrm{Cay}(T, S)$ is defined for a finite abelian group $T$ and an inverse-closed subset $S \subseteq T \setminus \{0\}$, with each vertex labeled by an element of $T$ and edges connecting pairs $u, v$ whenever $u-v \in S$. Such graphs are $|S|$-regular, undirected, and often highly symmetric. A graph is Ramanujan if all nontrivial adjacency eigenvalues $\lambda$ satisfy $|\lambda| \leq 2\sqrt{k-1}$, with $k=|S|$. In the abelian context, the eigenvalues are obtained explicitly as $\lambda_\chi = \sum_{s \in S} \chi(s)$ for each additive character $\chi$ of $T$ (Bibak et al., 2020).

Notable constructions include:

• Cayley graphs over $(\mathbb{Z}_p[i], S_2)$ with $S_2$ the set of norm one units, giving $(p+1)$-regular, abelian Ramanujan graphs for primes $p \equiv 3 \pmod{4}$ (Bibak et al., 2020).
• Unitary Cayley graphs $G_R = X(R, R^*)$ over finite commutative rings $R$, where $R^*$ is the group of units, yielding $|R^*|$-regular abelian graphs with explicit eigenvalue description (Podestá et al., 2020).
• Graphs from level sets of algebraic varieties in finite fields, e.g., norm one sets, hyperbolas, or cubic curves, e.g., $$N_2 = \{z\in\mathbb{F}_{q^2}^* \mid \mathrm{Norm}(z)=1\}$$ (Satake, 18 Jan 2026).

These graphs produce infinite families for certain classes of rings or fields and are often integral and explicitly analyzable via character theory.

2. Spectral Theory and Ramanujan Criteria

The spectrum of a Cayley graph $\mathrm{Cay}(T, S)$ on an abelian group $T$ is determined by the sums $\lambda_\chi = \sum_{s \in S} \chi(s)$, where $\chi$ ranges over the group of additive characters. The trivial character yields the top eigenvalue $k=|S|$, and Ramanujan-optimally requires that all non-principal eigenvalues satisfy $|\lambda_\chi| \leq 2\sqrt{k-1}$.

Sharp spectral bounds are obtained from deep results in algebraic geometry and analytic number theory:

• Deligne's bound gives $|\sum_{s \in S_2} \psi(s)| \leq 2\sqrt{p}$ for nontrivial additive characters $\psi$ when $S_2$ is the set of norm-one elements in $\mathbb{F}_{p^2}$ (Bibak et al., 2020).
• For certain algebraic sets $S$, such as norm-1 or quadratic forms, corresponding character sums are bounded by $2\sqrt{q}$ using the Weil or Hasse-Weil bounds (Satake, 18 Jan 2026).

In the context of unitary Cayley graphs over finite rings, explicit Ramanujan criteria are available: for a local ring $(R, m)$ of order $r$ and maximal ideal size $m$, the graph is Ramanujan precisely when $r=2m$, or $r \geq (m+1)^2$ and $m \neq 2$ (Podestá et al., 2020). For non-local rings, there is a complete classification of all abelian Ramanujan graphs that can be constructed in this manner.

3. Exemplary Infinite Families and Algebraic Frameworks

Several infinite or parametrized families of Abelian Ramanujan graphs are now characterized:

• Gaussian Integer Cayley Graphs: For each $p \equiv 3 \pmod{4}$, $\mathcal{G}_p = \mathrm{Cay}(\mathbb{Z}_p[i], S_2)$ forms a connected $(p+1)$-regular Ramanujan graph, with $S_2$ the set of Gaussian integers of norm $1$ modulo $p$ (Bibak et al., 2020).
• Norm-One Graphs over Finite Fields: For $q = p^k$, the graph $L_q = \mathrm{Cay}(\mathbb{F}_{q^2}^+, N_2)$, $$N_2 = \{z \in \mathbb{F}_{q^2}^* : \mathrm{Norm}(z)=1\}$$, is $(q+1)$-regular and Ramanujan (Satake, 18 Jan 2026).
• Unitary Cayley Graphs over Commutative Rings: Rings of “odd-type” and certain direct product structures yield equienergetic pairs and larger tuples of integral Ramanujan graphs (Podestá et al., 2020).

Such families are typically not bipartite and exhibit strong expansion, with explicit construction permitting detailed spectral computation and combinatorial analysis.

4. Eigenvalue Computation via Character Sums

For abelian Cayley graphs, all adjacency eigenvalues are determined by examining character sums over the generating set. For $S$ arising as an algebraic level set (e.g., the norm-one set), the character sums involved are often Kloosterman, Gauss, or more general exponential sums:

• For norm-one sets, Deligne-Weil bounds imply $|\lambda_\chi| \leq 2\sqrt{q}$ for all nontrivial $\chi$ (Bibak et al., 2020, Satake, 18 Jan 2026).
• For cubic curves, as in the almost-Ramanujan case, the Weil bound gives $|\lambda_\chi| \leq 2\sqrt{q}$ (Satake, 18 Jan 2026).
• In the ring setting, the Artin decomposition of the ring into local components reduces the spectral computation to manageable sums, allowing explicit tabulation of eigenvalues for various parameters (Podestá et al., 2020).

The tight spectral control explains the optimal expansion and related combinatorial properties.

5. Interplay with Coding Theory: Quasi-Perfect Lee Codes

There is a close connection between certain abelian Ramanujan graphs and constructions of 2-quasi-perfect Lee codes. Specifically, the Cayley graph obtained from an appropriate generator set $S$ over $(\mathbb{F}_{q^2}, +)$ is nearly Ramanujan, ensuring small spectral radius and thus small graph diameter, which via classical duality reduces the code's covering radius to 3. This construction produces infinite families of 2-quasi-perfect Lee codes whose parameters (length, dimension) are dictated by the size and structure of $S$ (Satake, 18 Jan 2026).

This relation is explicit for the Gaussian integer graphs and norm-one graphs, with the code constructions parallel to the spectral analysis via character sum bounds (Bibak et al., 2020, Satake, 18 Jan 2026).

6. Comparison with Non-Abelian Ramanujan Graphs

Classical non-abelian Ramanujan graphs, such as those from Lubotzky–Phillips–Sarnak or Margulis, arise via deep connections with quaternion algebras and non-abelian groups, achieving Ramanujan bounds for broader ranges of degrees. Abelian Ramanujan graphs are typically more elementary in construction and permit a full spectral description via character theory, but only achieve Ramanujan optimality for restricted values of degree and do not cover arbitrary degree expansion. Their universal covering is the infinite abelian Cayley-sum tree, in contrast to the richer geometric structure in the non-abelian setting (Podestá et al., 2020).

7. Limitations, Open Directions, and Context in Higher-Dimensional Abelian Varieties

Within isogeny graphs of principally polarized superspecial abelian varieties, Ramanujan properties are well understood only for dimension $g=1$; higher-dimensional analogues ($g > 1$) are typically not Ramanujan due to the existence of non-tempered automorphic representations and the absence of a general Riemann hypothesis for local Hecke algebras in these settings. Only a few sporadic two-vertex or small-parameter cases achieve Ramanujan optimality in higher genus, and no infinite families are known in this direction (Jordan, 2021, Jordan et al., 2020).

A plausible implication is that the pursuit of new infinite families of higher-rank abelian Ramanujan graphs may require fundamentally new techniques or different algebraic constructions.

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References:

• (Bibak et al., 2020) Bibak–Kapron–Srinivasan, The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs
• (Podestá et al., 2020) Podestá & Videla, Integral equienergetic non-isospectral unitary Cayley graphs
• (Satake, 18 Jan 2026) Forey–Fresán–Kowalski–Wigderson, 2-quasi-perfect Lee codes and abelian Ramanujan graphs: a new construction and relationship
• (Jordan, 2021) Jordan–Zaytman, Isogeny graphs of superspecial abelian varieties
• (Jordan et al., 2020) Eisenträger et al., Isogeny graphs of superspecial abelian varieties and Brandt matrices