## A strengthening and a multipartite generalization of the Alon-Boppana-Serre theorem

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**138**(2010), 3899-3909 Request permission

## Abstract:

The Alon-Boppana theorem confirms that for every $\varepsilon >0$ and every integer $d\ge 3$, there are only finitely many $d$-regular graphs whose second largest eigenvalue is at most $2\sqrt {d-1}-\varepsilon$. Serre gave a strengthening showing that a positive proportion of eigenvalues of any $d$-regular graph must be bigger than $2\sqrt {d-1}-\varepsilon$. We provide a multipartite version of this result. Our proofs are elementary and also work in the case when graphs are not regular. In the simplest, monopartite case, our result extends the Alon-Boppana-Serre result to non-regular graphs of minimum degree $d$ and bounded maximum degree. The two-partite result shows that for every $\varepsilon >0$ and any positive integers $d_1,d_2,d$, every $n$-vertex graph of maximum degree at most $d$, whose vertex set is the union of (not necessarily disjoint) subsets $V_1,V_2$, such that every vertex in $V_i$ has at least $d_i$ neighbors in $V_{3-i}$ for $i=1,2$, has $\Omega _\varepsilon (n)$ eigenvalues that are larger than $\sqrt {d_1-1}+\sqrt {d_2-1}-\varepsilon$. Finally, we strengthen the Alon-Boppana-Serre theorem by showing that the lower bound $2\sqrt {d-1}-\varepsilon$ can be replaced by $2\sqrt {d-1} + \delta$ for some $\delta >0$ if graphs have bounded “global girth”. On the other side of the spectrum, if the odd girth is large, then we get an Alon-Boppana-Serre type theorem for the negative eigenvalues as well.## References

- N. Alon,
*Eigenvalues and expanders*, Combinatorica**6**(1986), no. 2, 83–96. Theory of computing (Singer Island, Fla., 1984). MR**875835**, DOI 10.1007/BF02579166 - N. Alon and V. D. Milman,
*$\lambda _1,$ isoperimetric inequalities for graphs, and superconcentrators*, J. Combin. Theory Ser. B**38**(1985), no. 1, 73–88. MR**782626**, DOI 10.1016/0095-8956(85)90092-9 - P. Cartier,
*Harmonic analysis on trees*, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 419–424. MR**0338272** - Sebastian M. Cioabă,
*Eigenvalues of graphs and a simple proof of a theorem of Greenberg*, Linear Algebra Appl.**416**(2006), no. 2-3, 776–782. MR**2242462**, DOI 10.1016/j.laa.2005.12.020 - Giuliana Davidoff, Peter Sarnak, and Alain Valette,
*Elementary number theory, group theory, and Ramanujan graphs*, London Mathematical Society Student Texts, vol. 55, Cambridge University Press, Cambridge, 2003. MR**1989434**, DOI 10.1017/CBO9780511615825 - E. B. Dynkin and M. B. Maljutov,
*Random walk on groups with a finite number of generators*, Dokl. Akad. Nauk SSSR**137**(1961), 1042–1045 (Russian). MR**0131904** - Keqin Feng and Wen-Ch’ing Winnie Li,
*Spectra of hypergraphs and applications*, J. Number Theory**60**(1996), no. 1, 1–22. MR**1405722**, DOI 10.1006/jnth.1996.0109 - Joel Friedman,
*Some geometric aspects of graphs and their eigenfunctions*, Duke Math. J.**69**(1993), no. 3, 487–525. MR**1208809**, DOI 10.1215/S0012-7094-93-06921-9 - Chris Godsil and Gordon Royle,
*Algebraic graph theory*, Graduate Texts in Mathematics, vol. 207, Springer-Verlag, New York, 2001. MR**1829620**, DOI 10.1007/978-1-4613-0163-9 - Y. Greenberg, Spectra of graphs and their covering trees (in Hebrew), Ph.D. thesis, Hebrew University of Jerusalem, 1995.
- Shlomo Hoory,
*A lower bound on the spectral radius of the universal cover of a graph*, J. Combin. Theory Ser. B**93**(2005), no. 1, 33–43. MR**2102266**, DOI 10.1016/j.jctb.2004.06.001 - Shlomo Hoory, Nathan Linial, and Avi Wigderson,
*Expander graphs and their applications*, Bull. Amer. Math. Soc. (N.S.)**43**(2006), no. 4, 439–561. MR**2247919**, DOI 10.1090/S0273-0979-06-01126-8 - Roger A. Horn and Charles R. Johnson,
*Matrix analysis*, Cambridge University Press, Cambridge, 1985. MR**832183**, DOI 10.1017/CBO9780511810817 - Harry Kesten,
*Symmetric random walks on groups*, Trans. Amer. Math. Soc.**92**(1959), 336–354. MR**109367**, DOI 10.1090/S0002-9947-1959-0109367-6 - Alexander Lubotzky and Tatiana Nagnibeda,
*Not every uniform tree covers Ramanujan graphs*, J. Combin. Theory Ser. B**74**(1998), no. 2, 202–212. MR**1654133**, DOI 10.1006/jctb.1998.1843 - A. Lubotzky, R. Phillips, and P. Sarnak,
*Ramanujan graphs*, Combinatorica**8**(1988), no. 3, 261–277. MR**963118**, DOI 10.1007/BF02126799 - G. A. Margulis,
*Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators*, Problemy Peredachi Informatsii**24**(1988), no. 1, 51–60 (Russian); English transl., Problems Inform. Transmission**24**(1988), no. 1, 39–46. MR**939574** - Bojan Mohar and Wolfgang Woess,
*A survey on spectra of infinite graphs*, Bull. London Math. Soc.**21**(1989), no. 3, 209–234. MR**986363**, DOI 10.1112/blms/21.3.209 - Moshe Morgenstern,
*Existence and explicit constructions of $q+1$ regular Ramanujan graphs for every prime power $q$*, J. Combin. Theory Ser. B**62**(1994), no. 1, 44–62. MR**1290630**, DOI 10.1006/jctb.1994.1054 - A. Nilli,
*On the second eigenvalue of a graph*, Discrete Math.**91**(1991), no. 2, 207–210. MR**1124768**, DOI 10.1016/0012-365X(91)90112-F - A. Nilli,
*Tight estimates for eigenvalues of regular graphs*, Electron. J. Combin.**11**(2004), no. 1, Note 9, 4. MR**2056091** - William L. Paschke,
*Lower bound for the norm of a vertex-transitive graph*, Math. Z.**213**(1993), no. 2, 225–239. MR**1221715**, DOI 10.1007/BF03025720 - Jean-Pierre Serre,
*Répartition asymptotique des valeurs propres de l’opérateur de Hecke $T_p$*, J. Amer. Math. Soc.**10**(1997), no. 1, 75–102 (French). MR**1396897**, DOI 10.1090/S0894-0347-97-00220-8 - Wolfgang Woess,
*Random walks on infinite graphs and groups*, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR**1743100**, DOI 10.1017/CBO9780511470967

## Additional Information

**Bojan Mohar**- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada
- MR Author ID: 126065
- ORCID: 0000-0002-7408-6148
- Email: mohar@sfu.ca
- Received by editor(s): February 4, 2010
- Published electronically: June 22, 2010
- Additional Notes: The author was supported in part by the Research Grant P1–0297 of ARRS (Slovenia), by an NSERC Discovery Grant (Canada) and by the Canada Research Chair program.

The author is on leave from IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. - Communicated by: Jim Haglund
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**138**(2010), 3899-3909 - MSC (2010): Primary 05C50
- DOI: https://doi.org/10.1090/S0002-9939-2010-10543-9
- MathSciNet review: 2679612