Expander graph (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Expander graph" in English language version.

Last modified: 17:21:11, 13 August 2026 (UTC)

Ref.Un. Ref.Website

Global rank English rank

3429doi.org

2^nd place

1512semanticscholar.org

15^th place

8^th place

1210huji.ac.il

3,705^th place

3,379^th place

99arxiv.org

49^th place

31^st place

66worldcat.org

4^th place

33adsabs.harvard.edu

14^th place

9^th place

22wiley.com

150^th place

174^th place

21princeton.edu

802^nd place

528^th place

21yale.edu

559^th place

389^th place

11jstor.org

23^rd place

15^th place

11weizmann.ac.il

6,932^nd place

9,400^th place

11quantamagazine.org

6,258^th place

4,046^th place

11psu.edu

299^th place

159^th place

11ams.org

434^th place

249^th place

adsabs.harvard.edu (Global: 14^th place; English: 9^th place)

ui.adsabs.harvard.edu

Puder, Doron (2015-08-21). "Expansion of random graphs: New proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. S2CID 253743928.
Puder, Doron (2015). "Expansion of random graphs: new proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. ISSN 0020-9910. S2CID 16411939.
Hoffman, A. J.; Howes, Leonard (1970). "On Eigenvalues and Colorings of Graphs, Ii". Annals of the New York Academy of Sciences. 175 (1): 238–242. Bibcode:1970NYASA.175..238H. doi:10.1111/j.1749-6632.1970.tb56474.x. ISSN 1749-6632. S2CID 85243045.

ams.org (Global: 434^th place; English: 249^th place)

Chung, F. R. K. (1989). "Diameters and eigenvalues". Journal of the American Mathematical Society. 2 (2): 187–196. doi:10.1090/S0894-0347-1989-0965008-X. ISSN 0894-0347.

arxiv.org (Global: 49^th place; English: 31^st place)

Friedman, Joel (2004-05-05). "A proof of Alon's second eigenvalue conjecture and related problems". arXiv:cs/0405020.
Puder, Doron (2015-08-21). "Expansion of random graphs: New proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. S2CID 253743928.
Puder, Doron (2015). "Expansion of random graphs: new proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. ISSN 0020-9910. S2CID 16411939.
Friedman, Joel; Puder, Doron (2023). "A note on the trace method for random regular graphs". Israel Journal of Mathematics. 256: 269–282. arXiv:2006.13605. doi:10.1007/s11856-023-2497-5. S2CID 220042379.
Huang, Jiaoyang; McKenzie, Theo; Yau, Horng-Tzer (2024). "Ramanujan Property and Edge Universality of Random Regular Graphs". arXiv:2412.20263 [math.PR].
Bilu, Yonatan; Linial, Nathan (2004-04-08). "Constructing expander graphs by 2-lifts and discrepancy vs. spectral gap". arXiv:math/0312022.
Michael B. Cohen (2016). Ramanujan Graphs in Polynomial Time. Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium. arXiv:1604.03544. doi:10.1109/FOCS.2016.37.
Hall, Chris; Puder, Doron; Sawin, William F. (2018). "Ramanujan coverings of graphs". Advances in Mathematics. 323: 367–410. arXiv:1506.02335. doi:10.1016/j.aim.2017.10.042.
Alexander, Clark (2021). "On Near Optimal Spectral Expander Graphs of Fixed Size". arXiv:2110.01407 [cs.DM].

doi.org (Global: 2^nd place; English: 2^nd place)

Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Definition 2.1 in Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Bobkov, Houdré & Tetali (2000) Bobkov, S.; Houdré, C.; Tetali, P. (2000), "λ_∞, vertex isoperimetry and concentration", Combinatorica, 20 (2): 153–172, doi:10.1007/s004930070018, S2CID 1173532.
Alon & Capalbo (2002) Alon, N.; Capalbo, M. (2002), "Explicit unique-neighbor expanders", The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings, p. 73, doi:10.1109/SFCS.2002.1181884, ISBN 978-0-7695-1822-0, S2CID 6364755
cf. Section 2.3 in Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
This definition of the spectral gap is from Section 2.3 in Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Hoory, Linial & Wigderson 2006, Definition 2.2. Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Dodziuk 1984. Dodziuk, Jozef (1984), "Difference equations, isoperimetric inequality and transience of certain random walks", Trans. Amer. Math. Soc., 284 (2): 787–794, doi:10.2307/1999107, JSTOR 1999107.
Theorem 2.4 in Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
See Theorem 1 and p.156, l.1 in Bobkov, Houdré & Tetali (2000). Note that $λ 2$ there corresponds to $2(d - λ 2)$ of the current article (see p.153, l.5) Bobkov, S.; Houdré, C.; Tetali, P. (2000), "λ_∞, vertex isoperimetry and concentration", Combinatorica, 20 (2): 153–172, doi:10.1007/s004930070018, S2CID 1173532.
see, e.g., Yehudayoff (2012) Yehudayoff, Amir (2012), "Proving expansion in three steps", ACM SIGACT News, 43 (3): 67–84, doi:10.1145/2421096.2421115, S2CID 18098370
Alon, Noga (1986). "Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory". Combinatorica. 6 (3): 207–219. doi:10.1007/BF02579382. S2CID 8666466.
see, e.g., p.9 of Goldreich (2011) Goldreich, Oded (2011), "Basic Facts about Expander Graphs", Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation (PDF), Lecture Notes in Computer Science (Vol. 6650), vol. 6650, pp. 451–464, doi:10.1007/978-3-642-22670-0_30, ISBN 978-3-642-22669-4
Theorem 2.7 of Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Definition 5.11 of Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Theorem 5.12 of Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Alon, Noga (1986-06-01). "Eigenvalues and expanders". Combinatorica. 6 (2): 83–96. doi:10.1007/BF02579166. ISSN 1439-6912. S2CID 41083612.
Theorem 7.10 of Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Puder, Doron (2015-08-21). "Expansion of random graphs: New proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. S2CID 253743928.
Puder, Doron (2015). "Expansion of random graphs: new proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. ISSN 0020-9910. S2CID 16411939.
Friedman, Joel; Puder, Doron (2023). "A note on the trace method for random regular graphs". Israel Journal of Mathematics. 256: 269–282. arXiv:2006.13605. doi:10.1007/s11856-023-2497-5. S2CID 220042379.
Reingold, O.; Vadhan, S.; Wigderson, A. (2000). "Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors". Proceedings 41st Annual Symposium on Foundations of Computer Science. IEEE Comput. Soc. pp. 3–13. doi:10.1109/sfcs.2000.892006. ISBN 0-7695-0850-2. S2CID 420651.
Bilu, Yonatan; Linial, Nathan (2006). "Lifts, discrepancy and nearly optimal spectral gap". Combinatorica. 26 (5): 495–519. doi:10.1007/s00493-006-0029-7. ISSN 0209-9683. S2CID 14422668.
Michael B. Cohen (2016). Ramanujan Graphs in Polynomial Time. Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium. arXiv:1604.03544. doi:10.1109/FOCS.2016.37.
Hall, Chris; Puder, Doron; Sawin, William F. (2018). "Ramanujan coverings of graphs". Advances in Mathematics. 323: 367–410. arXiv:1506.02335. doi:10.1016/j.aim.2017.10.042.
Alon, N.; Roichman, Y. (1994). "Random Cayley graphs and Expanders". Random Structures and Algorithms. 5 (2). Wiley Online Library: 271–284. doi:10.1002/rsa.3240050203.
Hoffman, A. J.; Howes, Leonard (1970). "On Eigenvalues and Colorings of Graphs, Ii". Annals of the New York Academy of Sciences. 175 (1): 238–242. Bibcode:1970NYASA.175..238H. doi:10.1111/j.1749-6632.1970.tb56474.x. ISSN 1749-6632. S2CID 85243045.
Alon, Noga; Krivelevich, Michael; Sudakov, Benny (1999-09-01). "Coloring Graphs with Sparse Neighborhoods". Journal of Combinatorial Theory. Series B. 77 (1): 73–82. doi:10.1006/jctb.1999.1910. ISSN 0095-8956.
Chung, F. R. K. (1989). "Diameters and eigenvalues". Journal of the American Mathematical Society. 2 (2): 187–196. doi:10.1090/S0894-0347-1989-0965008-X. ISSN 0894-0347.

huji.ac.il (Global: 3,705^th place; English: 3,379^th place)

cs.huji.ac.il

Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Definition 2.1 in Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
cf. Section 2.3 in Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
This definition of the spectral gap is from Section 2.3 in Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Hoory, Linial & Wigderson 2006, Definition 2.2. Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Theorem 2.4 in Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Theorem 2.7 of Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Definition 5.11 of Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Theorem 5.12 of Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8
Theorem 7.10 of Hoory, Linial & Wigderson (2006) Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications" (PDF), Bulletin of the American Mathematical Society, New Series, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8

jstor.org (Global: 23^rd place; English: 15^th place)

Dodziuk 1984. Dodziuk, Jozef (1984), "Difference equations, isoperimetric inequality and transience of certain random walks", Trans. Amer. Math. Soc., 284 (2): 787–794, doi:10.2307/1999107, JSTOR 1999107.

princeton.edu (Global: 802^nd place; English: 528^th place)

annals.math.princeton.edu

Adam Marcus; Daniel Spielman; Nikhil Srivastava (2013). Interlacing families I: Bipartite Ramanujan graphs of all degrees (PDF). Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium.

psu.edu (Global: 299^th place; English: 159^th place)

citeseerx.ist.psu.edu

Pinkser, M. (1973). "On the Complexity of a Concentrator". SIAM Journal on Computing. SIAM. CiteSeerX 10.1.1.393.1430. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help)

quantamagazine.org (Global: 6,258^th place; English: 4,046^th place)

Sloman, Leila (2025-04-18). "New Proof Settles Decades-Old Bet About Connected Networks". Quanta Magazine. Retrieved 2025-05-06.

semanticscholar.org (Global: 15^th place; English: 8^th place)

api.semanticscholar.org

Bobkov, Houdré & Tetali (2000) Bobkov, S.; Houdré, C.; Tetali, P. (2000), "λ_∞, vertex isoperimetry and concentration", Combinatorica, 20 (2): 153–172, doi:10.1007/s004930070018, S2CID 1173532.
Alon & Capalbo (2002) Alon, N.; Capalbo, M. (2002), "Explicit unique-neighbor expanders", The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings, p. 73, doi:10.1109/SFCS.2002.1181884, ISBN 978-0-7695-1822-0, S2CID 6364755
See Theorem 1 and p.156, l.1 in Bobkov, Houdré & Tetali (2000). Note that $λ 2$ there corresponds to $2(d - λ 2)$ of the current article (see p.153, l.5) Bobkov, S.; Houdré, C.; Tetali, P. (2000), "λ_∞, vertex isoperimetry and concentration", Combinatorica, 20 (2): 153–172, doi:10.1007/s004930070018, S2CID 1173532.
see, e.g., Yehudayoff (2012) Yehudayoff, Amir (2012), "Proving expansion in three steps", ACM SIGACT News, 43 (3): 67–84, doi:10.1145/2421096.2421115, S2CID 18098370
Alon, Noga (1986). "Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory". Combinatorica. 6 (3): 207–219. doi:10.1007/BF02579382. S2CID 8666466.
Alon, Noga (1986-06-01). "Eigenvalues and expanders". Combinatorica. 6 (2): 83–96. doi:10.1007/BF02579166. ISSN 1439-6912. S2CID 41083612.
Puder, Doron (2015-08-21). "Expansion of random graphs: New proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. S2CID 253743928.
Puder, Doron (2015). "Expansion of random graphs: new proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. ISSN 0020-9910. S2CID 16411939.
Friedman, Joel; Puder, Doron (2023). "A note on the trace method for random regular graphs". Israel Journal of Mathematics. 256: 269–282. arXiv:2006.13605. doi:10.1007/s11856-023-2497-5. S2CID 220042379.
Reingold, O.; Vadhan, S.; Wigderson, A. (2000). "Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors". Proceedings 41st Annual Symposium on Foundations of Computer Science. IEEE Comput. Soc. pp. 3–13. doi:10.1109/sfcs.2000.892006. ISBN 0-7695-0850-2. S2CID 420651.
Bilu, Yonatan; Linial, Nathan (2006). "Lifts, discrepancy and nearly optimal spectral gap". Combinatorica. 26 (5): 495–519. doi:10.1007/s00493-006-0029-7. ISSN 0209-9683. S2CID 14422668.
Hoffman, A. J.; Howes, Leonard (1970). "On Eigenvalues and Colorings of Graphs, Ii". Annals of the New York Academy of Sciences. 175 (1): 238–242. Bibcode:1970NYASA.175..238H. doi:10.1111/j.1749-6632.1970.tb56474.x. ISSN 1749-6632. S2CID 85243045.

weizmann.ac.il (Global: 6,932^nd place; English: 9,400^th place)

wisdom.weizmann.ac.il

see, e.g., p.9 of Goldreich (2011) Goldreich, Oded (2011), "Basic Facts about Expander Graphs", Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation (PDF), Lecture Notes in Computer Science (Vol. 6650), vol. 6650, pp. 451–464, doi:10.1007/978-3-642-22670-0_30, ISBN 978-3-642-22669-4

wiley.com (Global: 150^th place; English: 174^th place)

onlinelibrary.wiley.com

Alon, N.; Roichman, Y. (1994). "Random Cayley graphs and Expanders". Random Structures and Algorithms. 5 (2). Wiley Online Library: 271–284. doi:10.1002/rsa.3240050203.
Hoffman, A. J.; Howes, Leonard (1970). "On Eigenvalues and Colorings of Graphs, Ii". Annals of the New York Academy of Sciences. 175 (1): 238–242. Bibcode:1970NYASA.175..238H. doi:10.1111/j.1749-6632.1970.tb56474.x. ISSN 1749-6632. S2CID 85243045.

worldcat.org (Global: 4^th place; English: 4^th place)

search.worldcat.org

Alon, Noga (1986-06-01). "Eigenvalues and expanders". Combinatorica. 6 (2): 83–96. doi:10.1007/BF02579166. ISSN 1439-6912. S2CID 41083612.
Puder, Doron (2015). "Expansion of random graphs: new proofs, new results". Inventiones Mathematicae. 201 (3): 845–908. arXiv:1212.5216. Bibcode:2015InMat.201..845P. doi:10.1007/s00222-014-0560-x. ISSN 0020-9910. S2CID 16411939.
Bilu, Yonatan; Linial, Nathan (2006). "Lifts, discrepancy and nearly optimal spectral gap". Combinatorica. 26 (5): 495–519. doi:10.1007/s00493-006-0029-7. ISSN 0209-9683. S2CID 14422668.
Hoffman, A. J.; Howes, Leonard (1970). "On Eigenvalues and Colorings of Graphs, Ii". Annals of the New York Academy of Sciences. 175 (1): 238–242. Bibcode:1970NYASA.175..238H. doi:10.1111/j.1749-6632.1970.tb56474.x. ISSN 1749-6632. S2CID 85243045.
Alon, Noga; Krivelevich, Michael; Sudakov, Benny (1999-09-01). "Coloring Graphs with Sparse Neighborhoods". Journal of Combinatorial Theory. Series B. 77 (1): 73–82. doi:10.1006/jctb.1999.1910. ISSN 0095-8956.
Chung, F. R. K. (1989). "Diameters and eigenvalues". Journal of the American Mathematical Society. 2 (2): 187–196. doi:10.1090/S0894-0347-1989-0965008-X. ISSN 0894-0347.

yale.edu (Global: 559^th place; English: 389^th place)

cs.yale.edu

Adam Marcus; Daniel Spielman; Nikhil Srivastava (2015). Interlacing families IV: Bipartite Ramanujan graphs of all sizes (PDF). Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium.

Expander graph (English Wikipedia)

adsabs.harvard.edu (Global: 14th place; English: 9th place)

ui.adsabs.harvard.edu

ams.org (Global: 434th place; English: 249th place)

arxiv.org (Global: 49th place; English: 31st place)

doi.org (Global: 2nd place; English: 2nd place)

huji.ac.il (Global: 3,705th place; English: 3,379th place)