63 research outputs found
Erratum: Percolation on random Johnson-Mehl tessellations and related models
We correct a simple error in Percolation on random Johnson-Mehl tessellations
and related models, Probability Theory and Related Fields 140 (2008), 417-468.
(See also arXiv:math/0610716)Comment: 7 page
A short proof of the Harris-Kesten Theorem
We give a short proof of the fundamental result that the critical probability
for bond percolation in the planar square lattice is equal to 1/2. The lower
bound was proved by Harris, who showed in 1960 that percolation does not occur
at . The other, more difficult, bound was proved by Kesten, who showed
in 1980 that percolation does occur for any .Comment: 17 pages, 9 figures; typos corrected. To appear in the Bulletin of
the London Mathematical Societ
Graphs and Hermitian matrices: exact interlacing
We prove conditions for equality between the extreme eigenvalues of a matrix
and its quotient. In particular, we give a lower bound on the largest singular
value of a matrix and generalize a result of Finck and Grohmann about the
largest eigenvalue of a graph
Large joints in graphs
We show that if G is a graph of sufficiently large order n containing as many
r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more
than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the
the r-partite Turan graph of order n. This structural result generalizes a
previous result that has been useful in extremal graph theory.Comment: 9 page
Walks and Paths in Trees
Recently Csikv\'ari \cite{csik} proved a conjecture of Nikiforov concerning
the number of closed walks on trees. Our aim is to extend his theorem to all
walks. In addition, we give a simpler proof of Csikv\'ari's result and answer
one of his questions in the negative. Finally we consider an analogous question
for paths rather than walks
An old approach to the giant component problem
In 1998, Molloy and Reed showed that, under suitable conditions, if a
sequence of degree sequences converges to a probability distribution , then
the size of the largest component in corresponding -vertex random graph is
asymptotically , where is a constant defined by the
solution to certain equations that can be interpreted as the survival
probability of a branching process associated to . There have been a number
of papers strengthening this result in various ways; here we prove a strong
form of the result (with exponential bounds on the probability of large
deviations) under minimal conditions.Comment: 24 pages; only minor change
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