## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Expanding graphs, Ramanujan graphs, and 1-factor perturbations

#### Abstract

We construct $(k \pm 1)$-regular graphs which provide sequences of expanders by adding or substracting appropriate 1-factors from given sequences of $k$-regular graphs. We compute numerical examples in a few cases for which the given sequences are from the work of Lubotzky, Phillips, and Sarnak (with $k-1$ the order of a finite field). If $k+1 = 7$, our construction results in a sequence of $7$-regular expanders with all spectral gaps at least $6 - 2\sqrt 5 \approx 1.52$; the corresponding minoration for a sequence of Ramanujan $7$-regular graphs (which is not known to exist) would be $7 - 2\sqrt 6 \approx 2.10$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 4 (2006), 673-680.

Dates
First available in Project Euclid: 16 January 2007

https://projecteuclid.org/euclid.bbms/1168957343

Digital Object Identifier
doi:10.36045/bbms/1168957343

Mathematical Reviews number (MathSciNet)
MR2300623

Zentralblatt MATH identifier
1130.05038

Subjects
Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

#### Citation

de la Harpe, Pierre; Musitelli, Antoine. Expanding graphs, Ramanujan graphs, and 1-factor perturbations. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 673--680. doi:10.36045/bbms/1168957343. https://projecteuclid.org/euclid.bbms/1168957343