# Bipartite Unique Neighbour Expanders via Ramanujan Graphs

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Related Work

## 3. Preliminaries

#### 3.1. Expander Graphs

- Vertex expansion. $\left({G}_{n}\right)$ is a $(\delta ,\alpha )$-vertex expander if, for every n and any subset $S\subseteq {V}_{{G}_{n}}$, if $\left|S\right|\le \delta |{V}_{{G}_{N}}|$, we have that $|{N}_{{G}_{N}}\left(S\right)|\ge \alpha |S|$.
- Edge expansion. $\left({G}_{n}\right)$ is a $(\delta ,\alpha )$-edge expander if, for every n and any subset $S\subseteq {V}_{{G}_{n}}$, if $\left|S\right|\le \delta |{V}_{{G}_{N}}|$, we have that an $\alpha $-fraction, at least, of the edges with one endpoint in S have their other endpoint outside of S.
- Spectral expansion. We assume that $\left({G}_{n}\right)$ are all d-regular, and let ${A}_{n}$ be the adjacency operator associated with ${G}_{n}$, so ${A}_{n}$ is indexed by vertices of ${G}_{n}$ and ${\left({A}_{n}\right)}_{uv}$ counts how many edges there are between u and v in ${G}_{n}$. Let ${\lambda}_{1}\ge \dots \ge {\lambda}_{{V}_{n}}$ be its spectrum. It can be seen that ${\lambda}_{1}=d$. Then, $\left({G}_{n}\right)$ is a $\lambda $-spectral expander if for all n and $i\ne 1$ we have $|{\lambda}_{i}|\le \lambda $.
- Unique neighbour expansion. $\left({G}_{n}\right)$ is a $\delta $-unique neighbour expander if, for every n, any subset $S\subseteq {V}_{{G}_{n}}$, of size $\delta |{V}_{{G}_{N}}|$ at most, has a unique neighbour.

#### 3.2. Bipartite Ramanujan Graphs

- Trivial: $\lambda =\pm \sqrt{cd}$, with eigenvectors fixed on either sides, or $\lambda =0$;
- $\lambda \in [\sqrt{d-1}-\sqrt{c-1},\sqrt{d-1}+\sqrt{c-1}]$ are the nontrivial positive eigenvalues;
- $\lambda \in [-\sqrt{c-1}-\sqrt{d-1},\sqrt{c-1}-\sqrt{d-1}]$ are the nontrivial negative eigenvalues. It should be noted that, since the graph is bipartite, $\lambda $ is an eigenvalue if and only if $-\lambda $ is an eigenvalue.

**Theorem**

**3**

**.**For every prime power q, there exists an explicit construction of a $(q+1,{q}^{3}+1)$-biregular Ramanujan graph.

## 4. Vertex Expansion in Biregular Ramanujan Graphs

**Theorem**

**4.**

#### 4.1. Comparison to Known Bounds

**Claim 1**(Expander Mixing Lemma for Bipartite Ramanujan Graphs)

**.**

**Proof.**

**Lemma**

**1.**

**Corollary**

**1.**

**Proof.**

#### 4.2. Proof of Theorem 2

**Lemma**

**2.**

**Lemma**

**3.**

**Proof**

**of Theorem 2.**

**Lemma**

**4.**

**Lemma**

**5.**

**Proof.**

- $t=0$. We use the same methods and find that the characteristic polynomial is$${\lambda}^{2}+(c-1+d-1)\lambda +(c-1)(d-1)$$$${\lambda}_{1}=-(c-1)\phantom{\rule{2.em}{0ex}},\phantom{\rule{2.em}{0ex}}{\lambda}_{2}=-(d-1).$$Using the initial conditions (${p}_{0}\left(0\right)=c/(c-1)$ and ${p}_{1}\left(0\right)=-c$), we obtain$$\alpha \left(0\right)=\frac{c}{c-1}\phantom{\rule{2.em}{0ex}},\phantom{\rule{2.em}{0ex}}\beta \left(0\right)=0$$$$\begin{array}{cc}\hfill |{p}_{\ell}\left(0\right)|& =|\alpha \left(0\right){\lambda}_{1}^{\ell}+\beta \left(0\right){\lambda}_{2}^{\ell}|\hfill \\ & =\frac{c}{c-1}{(c-1)}^{\ell}\hfill \\ & <2l{(c-1)}^{\ell /2}{(c-1)}^{\ell /2}\hfill \\ & <2l{(c-1)}^{\ell /2}{(d-1)}^{\ell /2}.\hfill \end{array}$$
- $t=\sqrt{d-1}+\sqrt{c-1}$. Then, $x={t}^{2}={(\sqrt{d-1}+\sqrt{c-1})}^{2}=d-1+c-1+2\sqrt{d-1}\sqrt{c-1}$, and the characteristic polynomial has a single root of multiplicity 2, namely,$$\lambda =\frac{x-(c-1)-(d-1)}{2}=\sqrt{d-1}\sqrt{c-1}.$$The solution, therefore, takes the form$${p}_{n}\left(x\right)=(\alpha \left(x\right)+n\beta \left(x\right)){(c-1)}^{n/2}{(d-1)}^{n/2}.$$Using the initial values, we obtain$$\alpha \left(x\right)=\frac{c}{c-1}\phantom{\rule{2.em}{0ex}},\phantom{\rule{2.em}{0ex}}\beta \left(x\right)=\frac{x-c}{\sqrt{d-1}\sqrt{c-1}}-\frac{c}{c-1}=2+\frac{d-2}{\sqrt{d-1}\sqrt{c-1}}-\frac{c}{c-1}.$$$1<\frac{c}{c-1}\le 2$ so $\beta \left(x\right)\le \sqrt{d-1}+1$, and, in total, we obtain$$\begin{array}{cc}\hfill |{p}_{\ell}\left(x\right)|& ={\left|\alpha \left(x\right)+\ell \beta \left(x\right)\right|(c-1)}^{\ell /2}{(d-1)}^{\ell /2}\hfill \\ & \le \left(\left|\frac{1}{\ell}\xb7\frac{c}{c-1}\right|+\left|\beta \left(x\right)\right|\right)\ell {(c-1)}^{\ell /2}{(d-1)}^{\ell /2}\hfill \\ & \le \left(2+\sqrt{d-1}\right)\ell {(c-1)}^{\ell /2}{(d-1)}^{\ell /2}\hfill \end{array}$$
- $t=\sqrt{d-1}-\sqrt{c-1}$. We obtain $x={t}^{2}=d-1+c-1-2\sqrt{d-1}\sqrt{c-1}$, and the rest follows the same calculations as in the previous case.

**Proof**

**of Lemma 3.**

## 5. Random Gadget

**Lemma**

**6.**

**Proof**

**of Lemma 6.**

## 6. Construction

#### 6.1. Routed Product Definition

**Lemma**

**7.**

#### 6.2. Proof of Theorem 1

## 7. Future Work

**Question**

**1.**

**Question**

**2.**

**Question**

**3.**

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of Lemma 1.**

**Proof**

**of Lemma 4.**

**Proof**

**of Lemma 7.**

## References

- Alon, N.; Capalbo, M. Explicit unique-neighbor expanders. In Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, Vancouver, BC, Canada, 19 November 2002; pp. 73–79. [Google Scholar]
- Capalbo, M.; Reingold, O.; Vadhan, S.; Wigderson, A. Randomness conductors and constant-degree expansion beyond the degree/2 barrier. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, Montréal, QC, Canada, 19–21 May 2002; pp. 659–668. [Google Scholar]
- Sipser, M.; Spielman, D.A. Expander codes. IEEE Trans. Inf. Theory
**1996**, 42, 1710–1722. [Google Scholar] [CrossRef] - Lin, T.; Hsieh, M. c
^{3}-Locally Testable Codes from Lossless Expanders. In Proceedings of the IEEE International Symposium on Information Theory, ISIT 2022, Espoo, Finland, 26 June–1 July 2022; pp. 1175–1180. [Google Scholar] [CrossRef] - Hsieh, J.; McKenzie, T.; Mohanty, S.; Paredes, P. Explicit two-sided unique-neighbor expanders. arXiv
**2023**, arXiv:2302.01212. [Google Scholar] [CrossRef] - Cohen, I.; Roth, R.; Ta-Shma, A. HDX condensers. In Proceedings of the 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), Santa Cruz, CA, USA, 6–9 November 2023; pp. 1649–1664. [Google Scholar]
- Kopparty, S.; Ron-Zewi, N.; Saraf, S. Simple constructions of unique neighbor expanders from error-correcting codes. arXiv
**2023**, arXiv:2310.19149. [Google Scholar] - Golowich, L. New explicit constant-degree lossless expanders. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Alexandria, VA, USA, 7–10 January 2024; pp. 4963–4971. [Google Scholar]
- Kahale, N. Eigenvalues and expansion of regular graphs. J. ACM
**1995**, 42, 1091–1106. [Google Scholar] [CrossRef] - Kamber, A.; Kaufman, T. Combinatorics via closed orbits: Number theoretic Ramanujan graphs are not unique neighbor expanders. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, 20–24 June 2022; pp. 426–435. [Google Scholar]
- Applebaum, B.; Kachlon, E. Sampling graphs without forbidden subgraphs and unbalanced expanders with negligible error. In Proceedings of the 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), Baltimore, MD, USA, 9–12 November 2019; pp. 171–179. [Google Scholar]
- Dinur, I.; Sudan, M.; Wigderson, A. Robust local testability of tensor products of LDPC codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques; Springer: Berlin/Heidelberg, Germany, 2006; pp. 304–315. [Google Scholar]
- Ben-Sasson, E.; Viderman, M. Tensor products of weakly smooth codes are robust. Theory Comput.
**2009**, 5, 239–255. [Google Scholar] [CrossRef] - Panteleev, P.; Kalachev, G. Asymptotically good quantum and locally testable classical LDPC codes. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, 20–24 June 2022; pp. 375–388. [Google Scholar]
- Kalachev, G.; Panteleev, P. Two-sided robustly testable codes. arXiv
**2022**, arXiv:2206.09973. [Google Scholar] - Arora, S.; Leighton, F.T.; Maggs, B.M. On-line algorithms for path selection in a nonblocking network. SIAM J. Comput.
**1996**, 25, 600–625. [Google Scholar] [CrossRef] - Pippenger, N. Self-routing superconcentrators. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, San Diego, CA, USA, 16–18 May 1993; pp. 355–361. [Google Scholar]
- Peleg, D.; Upfal, E. The token distribution problem. SIAM J. Comput.
**1989**, 18, 229–243. [Google Scholar] [CrossRef] - Becker, O. Symmetric unique neighbor expanders and good LDPC codes. Discret. Appl. Math.
**2016**, 211, 211–216. [Google Scholar] [CrossRef] - Nilli, A. On the second eigenvalue of a graph. Discret. Math.
**1991**, 91, 207–210. [Google Scholar] [CrossRef] - Godsil, C.D.; Mohar, B. Walk generating functions and spectral measures of infinite graphs. Linear Algebra Its Appl.
**1988**, 107, 191–206. [Google Scholar] [CrossRef] - Spectra of Regular Graphs and Hypergraphs and Orthogonal Polynomials. Eur. J. Comb.
**1996**, 17, 461–477. [CrossRef] - Feng, K.; Li, W.C.W. Spectra of hypergraphs and applications. J. Number Theory
**1996**, 60, 1–22. [Google Scholar] [CrossRef] - Lubotzky, A.; Phillips, R.; Sarnak, P. Ramanujan graphs. Combinatorica
**1988**, 8, 261–277. [Google Scholar] [CrossRef] - Margulis, G.A. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Probl. Peredachi Informatsii
**1988**, 24, 51–60. [Google Scholar] - Marcus, A.; Spielman, D.A.; Srivastava, N. Interlacing families I: Bipartite Ramanujan graphs of all degrees. In Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, Berkeley, CA, USA, 26–29 October 2013; pp. 529–537. [Google Scholar]
- Gribinski, A.; Marcus, A.W. Existence and polynomial time construction of biregular, bipartite Ramanujan graphs of all degrees. arXiv
**2021**, arXiv:2108.02534. [Google Scholar] - Brito, G.; Dumitriu, I.; Harris, K.D. Spectral gap in random bipartite biregular graphs and applications. Comb. Probab. Comput.
**2022**, 31, 229–267. [Google Scholar] [CrossRef] - Ballantine, C.; Feigon, B.; Ganapathy, R.; Kool, J.; Maurischat, K.; Wooding, A. Explicit construction of Ramanujan bigraphs. In Women in Numbers Europe; Springer: Berlin/Heidelberg, Germany, 2015; pp. 1–16. [Google Scholar]
- Kamber, A. L
^{p}Expander Graphs. arXiv**2019**, arXiv:1609.04433. [Google Scholar] [CrossRef] - Pippenger, N. Superconcentrators. SIAM J. Comput.
**1977**, 6, 298–304. [Google Scholar] [CrossRef] - Vadhan, S.P. Pseudorandomness. In Foundations and Trends in Theoretical Computer Science; Now Publishers: Norwell, MA, USA, 2012; Volume 7. [Google Scholar]
- Tanner, R. A recursive approach to low complexity codes. IEEE Trans. Inf. Theory
**1981**, 27, 533–547. [Google Scholar] [CrossRef]

**Figure 1.**An example of a bipartite graph G (dashed, red), a small gadget ${G}_{0}$ (dotted, green), and the routed product ${G}^{\prime}=G\circ {G}_{0}$ (solid, blue). The set $S\subseteq L$ has a neighbour $v\in R$, and so S is associated with a set ${S}^{\prime}$ of left-side vertices of the copy of ${G}_{0}$ associated with v. Since $({i}^{\prime},j)$ is the only edge connecting j to ${S}^{\prime}$ in ${G}_{0}$, we have that $(v,j)$ is a unique neighbour of S in ${G}^{\prime}$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Asherov, R.; Dinur, I.
Bipartite Unique Neighbour Expanders via Ramanujan Graphs. *Entropy* **2024**, *26*, 348.
https://doi.org/10.3390/e26040348

**AMA Style**

Asherov R, Dinur I.
Bipartite Unique Neighbour Expanders via Ramanujan Graphs. *Entropy*. 2024; 26(4):348.
https://doi.org/10.3390/e26040348

**Chicago/Turabian Style**

Asherov, Ron, and Irit Dinur.
2024. "Bipartite Unique Neighbour Expanders via Ramanujan Graphs" *Entropy* 26, no. 4: 348.
https://doi.org/10.3390/e26040348