The First Eigenvalue of Random Graphs

  • Authors:

  • Svante Janson

  • Affiliations:

  • Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden (e-mail: svante.janson@math.uu.se URL http://www.math.uu.se/~svante/)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2005

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Abstract

We extend a result by Füredi and Komlós and show that the first eigenvalue of a random graph is asymptotically normal, both for $G_{n,p}$ and $G_{n,m}$, provided $np\geq n^\delta$ or $m/n\geq n^\delta$ for some $\delta0$. The asymptotic variance is of order $p$ for $G_{n,p}$, and $n^{-1}$ for $G_{n,m}$. This gives a (partial) solution to a problem raised by Krivelevich and Sudakov.The formula for the asymptotic mean involves a mysterious power series.