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Proof of Alon’s Second Eigenvalue Conjecture and Related Problems
ISBN-13:
9781470405168
Einband:
PDF
Seiten:
100
Autor:
Joel Friedman
eBook Typ:
PDF
eBook Format:
PDF
Kopierschutz:
Adobe DRM [Hard-DRM]
Sprache:
Deutsch
Beschreibung:
A $d$-regular graph has largest or first (adjacency matrix) eigenvalue $lambda_1=d$. Consider for an even $dge 4$, a random $d$-regular graph model formed from $d/2$ uniform, independent permutations on ${1,ldots,n}$. The author shows that for any $epsilon>0$ all eigenvalues aside from $lambda_1=d$ are bounded by $2sqrt{d-1};+epsilon$ with probability $1-O(n^{- au})$, where $ au=lceil bigl(sqrt{d-1};+1bigr)/2ceil-1$. He also shows that this probability is at most $1-c/n^{ au'}$, for a constant $c$ and a $ au'$ that is either $ au$ or $ au+1$ ("e;more often"e; $ au$ than $ au+1$). He proves related theorems for other models of random graphs, including models with $d$ odd.
A $d$-regular graph has largest or first (adjacency matrix) eigenvalue $lambda_1=d$. Consider for an even $dge 4$, a random $d$-regular graph model formed from $d/2$ uniform, independent permutations on ${1,ldots,n}$. The author shows that for any $epsilon>0$ all eigenvalues aside from $lambda_1=d$ are bounded by $2sqrt{d-1};+epsilon$ with probability $1-O(n^{- au})$, where $ au=lceil bigl(sqrt{d-1};+1bigr)/2ceil-1$. He also shows that this probability is at most $1-c/n^{ au'}$, for a constant $c$ and a $ au'$ that is either $ au$ or $ au+1$ ("e;more often"e; $ au$ than $ au+1$). He proves related theorems for other models of random graphs, including models with $d$ odd.