21 research outputs found
On the Spectrum of the Derangement Graph
We derive several interesting formulae for the eigenvalues of the derangement graph and use them to settle affirmatively a conjecture of Ku regarding the least eigenvalue
Cyclic decomposition of k-permutations and eigenvalues of the arrangement graphs
The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of
an n-element set as vertices where two k-permutations are adjacent if they
agree in exactly k-1 positions. We introduce a cyclic decomposition for
k-permutations and show that this gives rise to a very fine equitable partition
of A(n,k). This equitable partition can be employed to compute the complete set
of eigenvalues (of the adjacency matrix) of A(n,k). Consequently, we determine
the eigenvalues of A(n,k) for small values of k. Finally, we show that any
eigenvalue of the Johnson graph J(n,k) is an eigenvalue of A(n,k) and that -k
is the smallest eigenvalue of A(n,k) with multiplicity O(n^k) for fixed k.Comment: 18 pages. Revised version according to a referee suggestion
A Proof of the Cameron-Ku conjecture
A family of permutations A \subset S_n is said to be intersecting if any two
permutations in A agree at some point, i.e. for any \sigma, \pi \in A, there is
some i such that \sigma(i)=\pi(i). Deza and Frankl showed that for such a
family, |A| <= (n-1)!. Cameron and Ku showed that if equality holds then A =
{\sigma \in S_{n}: \sigma(i)=j} for some i and j. They conjectured a
`stability' version of this result, namely that there exists a constant c < 1
such that if A \subset S_{n} is an intersecting family of size at least
c(n-1)!, then there exist i and j such that every permutation in A maps i to j
(we call such a family `centred'). They also made the stronger `Hilton-Milner'
type conjecture that for n \geq 6, if A \subset S_{n} is a non-centred
intersecting family, then A cannot be larger than the family C = {\sigma \in
S_{n}: \sigma(1)=1, \sigma(i)=i \textrm{for some} i > 2} \cup {(12)}, which has
size (1-1/e+o(1))(n-1)!.
We prove the stability conjecture, and also the Hilton-Milner type conjecture
for n sufficiently large. Our proof makes use of the classical representation
theory of S_{n}. One of our key tools will be an extremal result on
cross-intersecting families of permutations, namely that for n \geq 4, if A,B
\subset S_{n} are cross-intersecting, then |A||B| \leq ((n-1)!)^{2}. This was a
conjecture of Leader; it was recently proved for n sufficiently large by
Friedgut, Pilpel and the author.Comment: Updated version with an expanded open problems sectio