One-Sided Expanders

• One-sided expanders are graph- and complex-theoretic objects that impose expansion in only one direction, allowing pseudorandom properties alongside phenomena forbidden by classical two-sided expansion.
• Their analysis uses spectral and combinatorial methods that reveal a quantifiable trade-off between the positive and negative spectrum, enabling efficient algorithms for coloring and independent set approximation.
• These structures have practical applications in coding theory, quantum LDPC codes, and locally testable codes, while establishing limits on algorithmic approximations under relaxed spectral conditions.

Searching arXiv for papers on one-sided expanders and related notions. Searching for "one-sided expander spectral lossless local spectral". One-sided expanders are graph- and complex-theoretic objects in which expansion is imposed only on one side of a symmetry that is usually controlled in two-sided theories. In current usage, the term covers several related notions: regular graphs with bounded second eigenvalue but unrestricted negative spectrum; bipartite or regular graphs with lossless small-set vertex expansion required only from one side; and simplicial complexes whose links satisfy one-sided local spectral expansion. Across these settings, the common feature is that one retains enough pseudorandom structure for strong structural, algorithmic, and coding-theoretic consequences, while permitting phenomena that classical two-sided expansion forbids, including large-magnitude negative eigenvalues and large independent sets (Buhai et al., 4 Aug 2025, Hsieh et al., 21 Apr 2025, Kaufman et al., 2022).

1. Formal notions and the role of “one-sidedness”

The label “one-sided” is not attached to a single universal definition. It refers instead to a family of conditions in which expansion is enforced only in one direction: on the positive side of the spectrum, from one side of a bipartition, or across links of a simplicial complex without controlling the bottom of the spectrum.

Setting	Object	One-sided condition
Spectral graph expansion	$d$-regular graph $G$ with normalized adjacency $\tilde A=A/d$	$\lambda_2(\tilde A)\le \lambda$, with no restriction on negative eigenvalues
Lossless vertex expansion	$d$-regular graph $G=(V,E)$	For $|S|\le \eta|V|$, $|N(S)|\ge (1-\varepsilon)d|S|$
Bipartite lossless expansion	$(w_0,w_1)$-regular bipartite graph $\Xi=(V_0,V_1,E)$	For $G$0, $G$1
Local spectral expansion	simplicial complex $G$2	For every link graph $G$3, $G$4

For regular graphs, a $G$5-regular graph $G$6 with normalized adjacency matrix $G$7 and eigenvalues

$G$8

is a $G$9-one-sided expander when

$\tilde A=A/d$0

The corresponding two-sided condition typically requires

$\tilde A=A/d$1

This difference is substantive: two-sided expansion rules out large independent sets by Hoffman's bound, whereas one-sided expansion permits arbitrarily many large-magnitude negative eigenvalues and therefore large independent sets (Buhai et al., 4 Aug 2025).

In the lossless setting, one-sidedness is directional. For a $\tilde A=A/d$2-regular graph, one-sided $\tilde A=A/d$3-vertex expansion means that every sufficiently small vertex set $\tilde A=A/d$4 has nearly the maximum possible neighborhood size: $\tilde A=A/d$5 In the bipartite formulation, the same requirement is imposed only from $\tilde A=A/d$6 to $\tilde A=A/d$7. This immediately yields a unique-neighbor lower bound: $\tilde A=A/d$8 and analogous formulas appear in the bipartite setting (Hsieh et al., 21 Apr 2025, Lin et al., 2022).

For simplicial complexes, one-sidedness is local and spectral. A finite $\tilde A=A/d$9-dimensional simplicial complex $\lambda_2(\tilde A)\le \lambda$0 is a $\lambda_2(\tilde A)\le \lambda$1-one-sided local spectral expander if for every $\lambda_2(\tilde A)\le \lambda$2 and every $\lambda_2(\tilde A)\le \lambda$3-face $\lambda_2(\tilde A)\le \lambda$4, the second-largest eigenvalue of the normalized adjacency operator $\lambda_2(\tilde A)\le \lambda$5 of the link graph satisfies $\lambda_2(\tilde A)\le \lambda$6. No bound is required on the most negative eigenvalue of the link graph (Kaufman et al., 2022).

2. Spectral structure beyond two-sided expansion

A central development in the spectral theory of one-sided expanders is the identification of a quantitative relation between positive and negative parts of the spectrum. For a graph $\lambda_2(\tilde A)\le \lambda$7, the top-threshold rank and bottom-threshold rank at levels $\lambda_2(\tilde A)\le \lambda$8 are defined by

$\lambda_2(\tilde A)\le \lambda$9

Section 4 of "Finding Colorings in One-Sided Expanders" proves that for every graph $d$0 and every pair of thresholds $d$1,

$d$2

The abstract isolates a particularly transparent corollary: the number of negative eigenvalues smaller than $d$3 is at most $d$4 times the number of eigenvalues larger than $d$5. The same source emphasizes that this phenomenon had, to the best of the authors’ knowledge, not been observed before, and that the spectral trade-off alone is insufficient for the later algorithmic results (Buhai et al., 4 Aug 2025).

This perspective complements the standard one-sided spectral definition used in approximation algorithms. For a $d$6-regular graph $d$7, with random-walk matrix $d$8, one says that $d$9 is a $G=(V,E)$0-one-sided spectral expander if $G=(V,E)$1. By Cheeger’s inequality, this is equivalent to the statement that every $G=(V,E)$2 with $G=(V,E)$3 satisfies

$G=(V,E)$4

The same paper also formulates a weaker certificate based on small-set vertex expansion: $G=(V,E)$5 and studies the case $G=(V,E)$6 together with a Sum-of-Squares certificate of this fact (Bafna et al., 2024).

A persistent misconception is that one-sided spectral expansion is merely a relaxed two-sided condition with proportionally weaker consequences. The available results point to a sharper distinction. One-sided expansion still forces rapid mixing on the positive side and supports spectral-subspace methods, but it does not suppress the bottom of the spectrum; consequently, it remains compatible with large independent sets, planted colorings, and strong negative-eigenvalue structure that classical expander theory treats as obstructions (Buhai et al., 4 Aug 2025).

3. Coloring, independent sets, and vertex cover

The algorithmic study of one-sided spectral expanders has produced both positive results and sharp hardness boundaries. For $G=(V,E)$7-colorable regular one-sided expanders, "Finding Colorings in One-Sided Expanders" proves a polynomial-time algorithm that outputs either an independent set of relative size at least $G=(V,E)$8 or a proper $G=(V,E)$9-coloring for all but an $|S|\le \eta|V|$0 fraction of the vertices, where the $|S|\le \eta|V|$1 term tends to $|S|\le \eta|V|$2 with the second largest eigenvalue of the normalized adjacency matrix. The same work derives an efficient $|S|\le \eta|V|$3-factor approximation algorithm for VERTEX COVER in sufficiently strong regular one-sided expanders, improving over a previous $|S|\le \eta|V|$4-factor approximation for an unspecified constant $|S|\le \eta|V|$5. It also proposes a stratification of $|S|\le \eta|V|$6-COLORING by reversible, row-stochastic $|S|\le \eta|V|$7 matrices with zero diagonal and second eigenvalue at most $|S|\le \eta|V|$8: if the matrix has no repeated rows, the corresponding coloring problem can be solved on one-sided expanders in polynomial time; if it has repeated rows, then under the Unique Games Conjecture it is NP-hard even to recover an $|S|\le \eta|V|$9-fraction of the coloring. For balanced $|N(S)|\ge (1-\varepsilon)d|S|$0-colorable one-sided expanders, the same framework recovers three independent sets covering all but an $|N(S)|\ge (1-\varepsilon)d|S|$1 fraction of the vertices (Buhai et al., 4 Aug 2025).

Earlier approximation results already showed that one-sided spectral expansion suffices for nontrivial independent-set rounding. "Rounding Large Independent Sets on Expanders" gives a poly$|N(S)|\ge (1-\varepsilon)d|S|$2-time algorithm which, when $|N(S)|\ge (1-\varepsilon)d|S|$3 is $|N(S)|\ge (1-\varepsilon)d|S|$4-almost $|N(S)|\ge (1-\varepsilon)d|S|$5-colorable and satisfies $|N(S)|\ge (1-\varepsilon)d|S|$6, finds an independent set of size at least $|N(S)|\ge (1-\varepsilon)d|S|$7. More generally, for every $|N(S)|\ge (1-\varepsilon)d|S|$8, if $|N(S)|\ge (1-\varepsilon)d|S|$9 contains an independent set of size at least $(w_0,w_1)$0 and $(w_0,w_1)$1, then a polynomial-time algorithm returns an independent set of size at least $(w_0,w_1)$2. The same paper extends the approach to graphs with a degree-$(w_0,w_1)$3 SoS certificate of $(w_0,w_1)$4-SSVE and proves that, in time $(w_0,w_1)$5, one can find an independent set of size $(w_0,w_1)$6 provided the graph contains an independent set of size at least $(w_0,w_1)$7 (Bafna et al., 2024).

The positive results are accompanied by hardness phenomena that separate $(w_0,w_1)$8-colorability from $(w_0,w_1)$9-colorability. Assuming the Unique Games Conjecture, it is NP-hard to distinguish whether a regular $\Xi=(V_0,V_1,E)$0-vertex graph with $\Xi=(V_0,V_1,E)$1 is $\Xi=(V_0,V_1,E)$2-almost $\Xi=(V_0,V_1,E)$3-colorable or has no independent set of size at least $\Xi=(V_0,V_1,E)$4. This establishes that one-sided spectral expansion does not, by itself, support the same bottom-eigenspace-enumeration guarantees known in two-sided expanders for almost-$\Xi=(V_0,V_1,E)$5-colorable graphs when $\Xi=(V_0,V_1,E)$6 (Bafna et al., 2024).

At the technical level, the rounding algorithms rely on a clustering property rather than on the bottom eigenspace. In a one-sided spectral expander with $\Xi=(V_0,V_1,E)$7, any three independent sets of density at least $\Xi=(V_0,V_1,E)$8 must contain a pair whose intersection has size at least $\Xi=(V_0,V_1,E)$9. The result is formalized in low-degree Sum-of-Squares and combined with global-correlation reduction via Raghavendra–Tan conditioning (Bafna et al., 2024). This suggests that one-sided expansion supports a different algorithmic mechanism from the classical Alon–Kahale paradigm.

4. Lossless vertex expansion and unique-neighbor variants

In the lossless regime, one-sided expansion is formulated combinatorially rather than spectrally. For a $G$00-regular graph $G$01, one-sided $G$02-vertex expansion requires the existence of a constant $G$03, depending only on $G$04, such that every subset $G$05 with $G$06 satisfies

$G$07

When $G$08 as $G$09, the graph is called a lossless vertex expander. "Explicit Lossless Vertex Expanders" proves the first explicit constant-degree construction of such objects: for every $G$10 there exists $G$11 such that for every integer $G$12 there is an explicit, polynomial-time constructible infinite family of $G$13-regular $G$14-vertex graphs satisfying the displayed inequality for all $G$15. These graphs admit a free action by a group of size $G$16, and the construction yields new families of quantum LDPC codes with linear-time decoding (Hsieh et al., 21 Apr 2025).

A bipartite version had already been used in the construction of $G$17-locally testable codes. For a $G$18-regular bipartite graph $G$19, one-sided $G$20-lossless expansion from $G$21 to $G$22 means

$G$23

The existence theorem quoted there states that for every aspect-ratio parameter $G$24, every sufficiently small $G$25, and every large $G$26, there exist integers $G$27 and a constant $G$28 such that one can explicitly construct an infinite family of $G$29-regular bipartite graphs with $G$30, carrying a free action of a finite group $G$31 with $G$32, and satisfying the one-sided lossless inequality (Lin et al., 2022).

A separate line of work focuses on simpler explicit constructions and parameter trade-offs. "New Explicit Constant-Degree Lossless Expanders" constructs explicit bipartite one-sided $G$33-lossless expanders with left-degree

$G$34

for arbitrary constant ratio $G$35. The paper compares this with the earlier explicit construction of Capalbo–Reingold–Vadhan–Wigderson, which had $G$36 and $G$37, and analyzes the new family by elementary counting together with the expander mixing lemma (Golowich, 2023).

Unique-neighbor expansion refines losslessness by counting right-vertices that are adjacent to exactly one selected left-vertex. For a $G$38-biregular bipartite graph $G$39, a left-$G$40 unique-neighbor expander satisfies

$G$41

"Unique-neighbor Expanders with Better Expansion for Polynomial-sized Sets" gives, among other results, the first explicit family of one-sided lossless expanders with unique-neighbor expansion for polynomial-sized sets from the other side and constant aspect ratio: graphs that are left-$G$42 lossless, right-$G$43 unique-neighbor, and right-$G$44 lossless (Chen, 2024).

5. High-dimensional one-sided expansion

One-sided expansion also appears in high-dimensional expanders, where the ambient objects are simplicial complexes rather than graphs. For a finite $G$45-dimensional simplicial complex $G$46, and for every $G$47 and $G$48-face $G$49, the link $G$50 has an underlying graph $G$51 with normalized adjacency operator $G$52. The condition

$G$53

for all such links defines a $G$54-one-sided local spectral expander (Kaufman et al., 2022).

Within this setting, Kaufman and Mass introduced the notion of double-balanced sets. If $G$55 has indicator $G$56, and for an $G$57-face $G$58 one defines the localization $G$59 and the restricted local functions $G$60, then $G$61 is $G$62-double balanced in dimension $G$63 when

$G$64

If this holds for every $G$65, the set is called $G$66-double balanced (Kaufman et al., 2022).

The main combinatorial consequence is a unique-neighbor-like expansion theorem for the operator $G$67, where $G$68 consists of those $G$69-faces containing exactly one $G$70-face from $G$71. If $G$72 is a $G$73-dimensional $G$74-one-sided local spectral expander, $G$75 is $G$76-double balanced, and

$G$77

then for sufficiently small $G$78,

$G$79

The proof uses decomposition of $G$80 by links, a Cheeger argument in each link, and a density-control step derived from double balance plus mixing (Kaufman et al., 2022).

These ideas feed into topological applications. If every nontrivial link is also a $G$81-coboundary expander, then every nontrivial $G$82-cocycle satisfies

$G$83

hence is $G$84-double balanced in dimension $G$85. Combining this with the $G$86-expansion theorem yields the lower bound

$G$87

for every nonzero $G$88-cohomology element, improving exponentially over the previous bound $G$89 (Kaufman et al., 2022).

6. Constructions, examples, applications, and limits

Several canonical constructions show that one-sided expanders arise naturally rather than as isolated exceptions. For one-sided spectral expansion in regular graphs, "Finding Colorings in One-Sided Expanders" identifies three model families: blow-ups of host graphs with $G$90 blocks and an $G$91-matrix of transition probabilities, where between-block graphs are biregular Ramanujan bipartite graphs; random planting in an expander, obtained by choosing a uniform random $G$92-coloring of a two-sided $G$93-regular expander and deleting monochromatic edges; and balanced $G$94-block graphs corresponding to the $G$95 matrix with zeros on the diagonal and $G$96 on each off-diagonal entry (Buhai et al., 4 Aug 2025).

The explicit lossless constructions are structurally different but analogous in spirit: a constant-size gadget is amplified through a highly expanding host. In (Hsieh et al., 21 Apr 2025), the graphs arise as a Tripartite Line Product of two incidence graphs built from Ramanujan Cayley cubical complexes and a random-like biregular gadget. In (Golowich, 2023), a constant-sized lossless-expander gadget is planted inside the neighborhoods of a larger bipartite spectral expander. In (Chen, 2024), tripartite product techniques are combined with biregular gadget existence arguments and a reduction from large girth or bicycle-freeness to vertex expansion.

The application profile is correspondingly broad. One-sided spectral expanders are used for coloring, independent set, and vertex-cover approximation, and the papers explicitly connect the relevant graph families to coding theory, PCP constructions, and the stochastic block model (Buhai et al., 4 Aug 2025). One-sided and two-sided lossless vertex expanders are used to construct locally testable codes and quantum LDPC codes, with several results emphasizing free group actions and linear-time or almost linear-time decoding (Lin et al., 2022, Hsieh et al., 21 Apr 2025, Chen, 2024). In high dimensions, one-sided local spectral expansion supports unique-neighbor-like expansion and lower bounds on the minimal distance of cohomology classes (Kaufman et al., 2022).

The present limits are also clear. One-sided spectral expansion does not collapse the distinction between easy and hard planted-coloring problems: almost-$G$97-colorability admits positive algorithms, whereas almost-$G$98-colorability remains Unique-Games-hard even when $G$99 (Bafna et al., 2024). In the $\tilde A=A/d$00-coloring framework, repeated rows in the governing matrix mark the hard side of the dichotomy (Buhai et al., 4 Aug 2025). In the lossless-expander literature, explicit two-sided guarantees were long unavailable at constant degree, and recent work treats their construction as a substantive advance rather than a routine extension (Hsieh et al., 21 Apr 2025).

A plausible implication is that “one-sided expansion” is best understood not as a weakened form of classical expansion but as a separate regime. It interpolates between planted-partition models, lossless small-set expansion, and local spectral expansion in complexes, while preserving enough structure for sharp algorithmic and coding-theoretic theorems and allowing negative-spectrum phenomena that are invisible in the classical two-sided framework.