Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T18:29:08.333Z Has data issue: false hasContentIssue false
  • English
  • Français

The Largest Eigenvalue of Sparse Random Graphs

Published online by Cambridge University Press:  28 January 2003

MICHAEL KRIVELEVICH  and
BENNY SUDAKOV
MICHAEL KRIVELEVICH
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: krivelev@math.tau.ac.il)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA and Institute for Advanced Study, Princeton, NJ 08540, USA (e-mail: bsudakov@math.princeton.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that, for all values of the edge probability $p(n)$, the largest eigenvalue of the random graph $G(n, p)$ satisfies almost surely $\lambda_1(G)=(1+o(1))\max\{\sqrt{\Delta}, np\}$, where Δ is the maximum degree of $G$, and the o(1) term tends to zero as $\max\{\sqrt{\Delta}, np\}$ tends to infinity.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 12 , Issue 1 , January 2003 , pp. 61 - 72
Copyright
2003 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)