1,327 research outputs found
The Zeta Function of a Hypergraph
We generalize the Ihara-Selberg zeta function to hypergraphs in a natural
way. Hashimoto's factorization results for biregular bipartite graphs apply,
leading to exact factorizations. For -regular hypergraphs, we show that
a modified Riemann hypothesis is true if and only if the hypergraph is
Ramanujan in the sense of Winnie Li and Patrick Sol\'e. Finally, we give an
example to show how the generalized zeta function can be applied to graphs to
distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.Comment: 24 pages, 6 figure
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
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