The Shannon capacity of a graph and the independence numbers of its powers

@article{Alon2006TheSC,
  title={The Shannon capacity of a graph and the independence numbers of its powers},
  author={Noga Alon and Eyal Lubetzky},
  journal={IEEE Transactions on Information Theory},
  year={2006},
  volume={52},
  pages={2172-2176}
}
  • N. Alon, Eyal Lubetzky
  • Published 1 May 2006
  • Mathematics, Computer Science
  • IEEE Transactions on Information Theory
The independence numbers of powers of graphs have been long studied, under several definitions of graph products, and in particular, under the strong graph product. We show that the series of independence numbers in strong powers of a fixed graph can exhibit a complex structure, implying that the Shannon capacity of a graph cannot be approximated (up to a subpolynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the series. This is true even if this prefix… 
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References

SHOWING 1-10 OF 24 REFERENCES
On the Shannon capacity of a graph
  • L. Lovász
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1979
TLDR
It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.
The Shannon Capacity of a Union
  • N. Alon
  • Mathematics, Computer Science
    Comb.
  • 1998
TLDR
It is shown that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities, disproves a conjecture of Shannon raised in 1956.
A limit theorem for the Shannon capacities of odd cycles. II
It follows from a construction for independent sets in the powers of odd cycles given in the predecessor of this paper that the limit as k goes to infinity of k + 1/2 - Θ(C 2k+1 ) is zero, where 6(G)
A nontrivial lower bound on the Shannon capacities of the complements of odd cycles
TLDR
A construction for independent sets in the powers of the complements of odd cycles shows that /spl alpha/(C~/sub 2n+3/(2/sup n/))/spl ges/2(2/Sup n/)+1.
Explicit Ramsey graphs and orthonormal labelings
  • N. Alon
  • Mathematics, Computer Science
    Electron. J. Comb.
  • 1994
TLDR
The results imply that the maximum possible Euclidean norm of a sum of n unit vectors in R n, so that among any three of them some two are orthogonal, is £(n 2=3 ).
Repeated communication and Ramsey graphs
TLDR
Studying the savings afforded by repeated use in two zero-error communication problems shows that some channels can communicate exponentially more bits in two uses than they can in one use, and that this is essentially the largest possible increase.
A comparison of the Delsarte and Lovász bounds
  • A. Schrijver
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1979
TLDR
Delsarte's linear programming bound is compared with Lov\acute{a}sz's \theta -function bound (an upper bound on the Shannon capacity of a graph) and the two bounds can be treated in a uniform fashion.
On Some Problems of Lovász Concerning the Shannon Capacity of a Graph
  • W. Haemers
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1979
The answers to several problems of Lov\hat{a}sz concerning the Shannon capacity of a graph are shown to be negative.
On majorization, factorization, and range inclusion of operators on Hilbert space
The purpose of this note is to show that a close relationship exists between the notions of majorization, factorization, and range inclusion for operators on a Hilbert space. Although fragments of
The Probabilistic Method
TLDR
A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
...
1
2
3
...