Ramanujan Graphs and Interlacing Families

Nikhil Srivastava

Ramanujan Graphs and Interlacing Families

T0 review · 0 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The interlacing-families method proves optimal expanders exist in every degree, the survey argues.

desk verdict A solid, honest survey of interlacing families and Ramanujan graphs, with no new results; the value is organizational and pedagogical, and it deserves review as a survey. read the letter →

arxiv 2412.20721 v1 pith:FQ7QVKXB submitted 2024-12-30 math.CO cs.DM

Nikhil Srivastava This is my paper · ORCID

classification math.COcs.DM MSC 05C5005C8046L54

keywords Ramanujangraphsinterlacingfamiliesspectralgraphtheoryexpectedcharacteristicpolynomialsmatchingrandomregularfreeprobabilitylifts

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This survey argues that one technique, interlacing families of polynomials, proves the existence of the best possible sparse expander graphs: for every degree $d$, bipartite $d$-regular Ramanujan graphs exist, and random $d$-regular graphs have positive probability of being one-sided Ramanujan. The technique replaces a random adjacency matrix by its expected characteristic polynomial and shows, through real-rootedness and interlacing, that the roots of that polynomial are attained as eigenvalue bounds by at least one realization. The survey also explains how the same idea extends from 2-lifts to $n$-lifts and to signings by group representations, and how the random-regular case connects to free probability through a polynomial convolution. A sympathetic reader should come away seeing the interlacing-families method as a unified explanation for results that were previously reached by number theory or by high-probability random matrix arguments.

What carries the argument

The central object is the expected characteristic polynomial of a random matrix, together with the interlacing family generated by conditioning on the random choices one at a time. An interlacing family is a collection of real-rooted polynomials whose averages remain real-rooted and whose roots bracket the roots of the average, so a root bound on the expected polynomial implies positive probability of the same bound for an individual realization. The load-bearing identities are: the matching-polynomial formula for the expected characteristic polynomial of a random signing; the matching-polynomial root bound of $2\sqrt{d-1}$ for graphs of maximum degree $d$; and, for random regular graphs, a polynomial-convolution identity that identifies the expected characteristic polynomial (after removing the trivial eigenvalue) with a repeated convolution of a two-point polynomial. The support of the corresponding free convolution of Bernoulli measures is the limiting spectral distribution of random $d$-regular graphs, which lives in $[-2\sqrt{d-1}, 2\sqrt{d-1}]$.

What would settle it

Find a single $d$-regular bipartite graph for which exhaustive search over all signings shows that every signing has spectral norm strictly greater than $2\sqrt{d-1}$; that would disprove Theorem 3.1 and the survey's central existence claim. For the random regular graph result, compute the second largest root of the expected characteristic polynomial for small $d$ and even $n$; if it ever exceeds $2\sqrt{d-1}$, the proof chain in Section 4 would be broken.

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Extended reading notes

Core claim

The survey's central claim is that the interlacing families method establishes eigenvalue bounds that were previously out of reach. Concretely, it reports two theorems: every $d$-regular bipartite graph has a signing whose spectral norm is at most $2\sqrt{d-1}$ (Theorem 3.1), and a random $d$-regular graph on $n$ vertices satisfies $\mathbb{P}[\lambda_2(A_G) \le 2\sqrt{d-1}] > 0$ (Theorem 4.1). The mechanism, stated as Theorem 2.1, is that for these random matrix models the expected characteristic polynomial $p_A(z) = \mathbb{E}\det(zI - A)$ is real-rooted and its $i$-th root $\lambda_i(p_A)$ satisfies $\mathbb{P}[\lambda_i(A) \le \lambda_i(p_A)] > 0$, so a root bound on the expectation becomes an existence statement for a single realization. The survey further reports that the same scheme works for $n$-covers of bipartite base graphs and for signings by group representations whose exterior powers are irreducible, and that the needed root bounds come from matching polynomials and from finite and free convolution.

Load-bearing premise

The survey assumes, without reproducing the proofs, that the cited interlacing families theorem and the cited root-bound identities for matching polynomials and their representation-theoretic generalization are correct; if any of those cited results fails, the surveyed existence theorems do not follow.

Editorial extensions

If this is right

Every $d$-regular bipartite graph has a signing, equivalently a 2-lift, whose new eigenvalues all lie in the Ramanujan interval, so iterating the signing step produces infinite families of bipartite Ramanujan graphs for every $d \ge 3$.
For every $d \ge 3$ and every even $n$, there exists a $d$-regular multigraph whose second eigenvalue is at most $2\sqrt{d-1}$; the bipartite version gives a two-sided Ramanujan graph of every such size.
The method yields a positive-probability guarantee rather than a high-probability one: it shows the desired graph exists inside the random model but does not by itself show that a random draw is usually Ramanujan.
The same interlacing framework, with suitable root bounds, extends to $n$-covers of bipartite base graphs and to signings by group representations whose exterior powers are irreducible, and it narrows the original existence question to the non-bipartite case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The structure of the proof suggests that the main remaining gap, infinite sequences of non-bipartite Ramanujan graphs for all degrees, would close if an interlacing family could be built for non-bipartite base graphs; the survey presents no obstruction, only a missing ingredient.
The positivity conclusion is inherently non-quantitative; any future theorem giving a probability bounded away from zero for random regular graphs would need new tools, because the expected-polynomial method is designed to control one realization rather than the typical one.
The convolution identity used for random regular graphs suggests a testable program: replace the two-point measure with other finitely supported measures and check whether the interlacing root bound still holds, which would give Ramanujan-type spectral guarantees for other structured random matrix models.
One could numerically test the chain in Section 4 on small cases by computing the expected characteristic polynomial of a random $d$-regular graph and verifying that its second root stays at or below $2\sqrt{d-1}$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. This is a survey, not a research paper. Srivastava claims no new results and proves no new theorems; the content is a synthesis of the interlacing families literature, mostly the author's own earlier papers with Marcus and Spielman plus Hall-Puder-Sawin, Mohanty-O'Donnell, and the recent strong-convergence work. The reader's assessment is correct on all counts: novelty zero, soundness high, circularity not an issue.

What the paper does well: it's a clear, accurate map of a decade of work. The structure is sensible—random covers first, then random regular graphs—and the connections are honestly drawn. The Walsh-convolution-to-free-convolution route in Section 4 is nicely explained, and the open questions section is genuinely useful. The author also gives credit where it's due (Friedman, Bordenave, Puder, Huang-McKenzie-Yau, etc.). For a lecture companion, this does its job.

Where are the soft spots? First, it is a survey, so if you want proofs and new techniques, this isn't the place; every central theorem is deferred to the original papers. That's not a flaw in a survey, but it does mean the paper's value is pedagogical and organizational, not mathematical. Second, there are a few small editing issues: 'multiplicative context' in Section 1.2 should be 'multiplicative constant,' and in Theorem 1.4 'd = pk + 1' should be 'p^k + 1.' Also the sentence 'It is possible Theorem 2.1 generalizes' is a grammatical stumble. None of this affects the math.

The self-citation point: it's the author's own method, so citing Marcus-Spielman-Srivastava heavily is appropriate, and the cited results are published and peer-reviewed. The reader's weakest assumption—that the cited theorems are correct—is real but unavoidable in any survey; you can't reproduce every proof and still keep it readable. I checked the proof sketches, especially the free convolution part, and they're consistent with the literature.

Recommendation: send it to peer review as an invited survey; it deserves referee time. The right referee should check attributions and the proof sketch in Section 4, not expect new results. I'd bring it to a reading group for people entering the area.

Referee Report

0 major / 4 minor

Summary. This survey, based on a lecture at the 2024 ICBS, explains the interlacing families method and its applications to the existence of Ramanujan graphs. It covers the spectral definition and Alon–Boppana background, the interlacing families theorem, random covers and signings of fixed graphs (with emphasis on Marcus–Spielman–Srivastava and Hall–Puder–Sawin), the construction of one-sided Ramanujan multigraphs of every size via Walsh and free convolution, and six open questions. The paper claims no new theorems; its contribution is expository, presenting proof sketches and contextualizing the cited literature.

Significance. If the surveyed theorems are correct, this is a valuable authoritative survey: it is written by one of the originators of the interlacing families method and gives a coherent route through a literature that spans spectral graph theory, matching polynomials, representation theory, and free probability. I checked the central statements against my knowledge: Theorem 2.1, Theorem 3.1, Theorem 3.4, and Theorem 4.1 are all stated accurately, and the Walsh-convolution/free-convolution route described in Section 4 is mathematically sound. The survey appropriately outsources full proofs to the original papers; this is normal for a survey and does not undermine the exposition. The presentation of the Hall–Puder–Sawin representation-theoretic framework is especially useful. The paper contains no code or machine-checked artifacts, but none are expected for a survey of this type.

minor comments (4)

[Theorem 1.4, Section 1.1] The statement "Let d = pk + 1" appears to have a missing superscript: the classical LPS–Margulis condition for these constructions is of the form d = p^k + 1 (or d = q + 1 for q a prime power). As printed, the formula reads as d = p·k + 1, which is not the intended statement.
[Section 1.2, paragraph on Friedman–Kohler and Puder] "Puder [48] proved it up to a multiplicative context" should read "up to a multiplicative constant"; the current phrase is meaningless as printed.
[Section 3.2, paragraph on group signings] "a unitary representation of γ" should be "a unitary representation of Γ"; the symbol γ is not defined here.
[Footnote 6, Section 4] The monotonicity argument mentioned in footnote 6 is essential for passing from the roots of p to the roots of χ[M](z); since it is the only step beyond the cited theorem, a sentence in the main text explaining it would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the survey relies on published, independently established theorems rather than deriving its conclusions from its own inputs.

full rationale

This is an expository survey of the interlacing families method. It establishes no new theorems; every load-bearing statement is either quoted from the literature or presented with an explicit pointer to an external proof. For example, Theorem 2.1 is stated after 'The following theorem follows from results in [40,42,28]' and the proof is deferred to those papers. The Section 4 proof of Theorem 4.1 uses the Walsh-convolution identity (4.3), attributed to [42,28], and the root-location bound Theorem 4.2, attributed to [43], both of which are published results with proofs independent of this survey. The free convolution of Bernoulli measures being the Kesten-McKay law is cited to McKay. Self-citation is abundant (Srivastava is a coauthor of the papers carrying the main theorems), but per the hard rules, citation to peer-reviewed, externally verifiable results is not a circular reduction: the survey does not define its objects in terms of the conclusion, fit any parameter to the target quantity, or invoke a uniqueness theorem to forbid alternatives. The only omissions are the absence of reproduced proofs of the cited theorems, which is normal in a survey and does not constitute circularity. The typo 'd = pk + 1' for what is presumably 'p^k + 1' in Theorem 1.4 is a presentation error, not a logical loop. No circular step can be exhibited from the paper's own equations, so the score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No new parameters or entities are introduced; the exposition relies on published theorems listed above.

assumptions (6)

standard math Theorem 2.1 (Interlacing Families): for random graph models R1' and R2, p_A is real-rooted and P[lambda_i(A) <= lambda_i(p_A)] > 0 for every i.
Invoked in Section 2 as the central engine of the method; proof referred to [40,42,28].
standard math Godsil-Gutman theorem: E det(zI - A_H,s) = M_H(z), the matching polynomial of H.
Used in Section 3.1 (Theorem 3.2) to connect random signings to matching polynomials.
standard math Heilmann-Lieb theorem: the matching polynomial of a graph of maximum degree d is real-rooted with all roots in [-2 sqrt(d-1), 2 sqrt(d-1)].
Used in Section 3.1 (Theorem 3.3) to bound the roots of the expected characteristic polynomial.
standard math Hall-Puder-Sawin identity (3.2): under property (P1), E det(zI - A_H,s) = E_{G ~ Cov_{n-1}} M_G(z); property (P2) yields an interlacing family.
Used in Section 3.2 (Theorems 3.4 and 3.5) to extend the result to n-covers and group signings.
standard math Walsh convolution preserves real-rootedness, and Theorem 4.2 bounds the largest root of a Walsh convolution by the support of Voiculescu's free convolution.
Used in Section 4 to prove the one-sided Ramanujan result for random d-regular graphs.
standard math Alon-Boppana bound: every d-regular graph has a nontrivial eigenvalue of size at least 2 sqrt(d-1) - o_n(1).
Used in Section 1.1 to motivate the Ramanujan threshold.

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Cite this review

Pith. "Pith review of Ramanujan Graphs and Interlacing Families." pith.science (2026). https://pith.science/paper/FQ7QVKXB

@misc{pith2026241220721,
  author       = {Pith},
  title        = {Pith review of: Ramanujan Graphs and Interlacing Families},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FQ7QVKXB}},
  note         = {Machine review of arXiv:2412.20721}
}

read the original abstract

This survey accompanies a lecture on the paper ``Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees'' by A. Marcus, D. Spielman, and N. Srivastava at the 2024 International Congress of Basic Science (ICBS) in July, 2024. Its purpose is to explain the developments surrounding this work over the past ten or so years, with an emphasis on connections to other areas of mathematics. Earlier surveys about the interlacing families method by the same authors focused on applications in functional analysis, whereas the focus here is on applications in spectral graph theory.

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