156 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
On the extreme eigenvalues of regular graphs
In this paper, we present an elementary proof of a theorem of Serre
concerning the greatest eigenvalues of -regular graphs. We also prove an
analogue of Serre's theorem regarding the least eigenvalues of -regular
graphs: given , there exist a positive constant
and a nonnegative integer such that for any -regular graph
with no odd cycles of length less than , the number of eigenvalues
of such that is at least . This
implies a result of Winnie Li.Comment: accepted to J.Combin.Theory, Series B. added 5 new references, some
comments on the constant c in Section
Constructing cospectral hypergraphs
Spectral hypergraph theory mainly concerns using hypergraph spectra to obtain
structural information about the given hypergraphs. The study of cospectral
hypergraphs is important since it reveals which hypergraph properties cannot be
deduced from their spectra. In this paper, we show a new method for
constructing cospectral uniform hypergraphs using two well-known hypergraph
representations: adjacency tensors and adjacency matrices
A Spectral Moore Bound for Bipartite Semiregular Graphs
Let be the maximum number of vertices of valency in a
-semiregular bipartite graph with second largest eigenvalue .
We obtain an upper bound for for . This bound is tight when there exists a distance-biregular
graph with particular parameters, and we develop the necessary properties of
distance-biregular graphs to prove this.Comment: 20 page
Global eigenvalue fluctuations of random biregular bipartite graphs
We compute the eigenvalue fluctuations of uniformly distributed random
biregular bipartite graphs with fixed and growing degrees for a large class of
analytic functions. As a key step in the proof, we obtain a total variation
distance bound for the Poisson approximation of the number of cycles and
cyclically non-backtracking walks in random biregular bipartite graphs, which
might be of independent interest. As an application, we translate the results
to adjacency matrices of uniformly distributed random regular hypergraphs.Comment: 45 pages, 5 figure
Tensor join of hypergraphs and its spectra
In this paper, we introduce three operations on hypergraphs by using tensors. We show that these three formulations are equivalent and we commonly call them as the tensor join. We show that any hypergraph can be viewed as a tensor join of hypergraphs. Tensor join enable us to obtain several existing and new classes of operations on hypergraphs. We compute the adjacency, the Laplacian, the normalized Laplacian spectrum of weighted hypergraphs constructed by this tensor join. Also we deduce some results on the spectra of hypergraphs in the literature. As an application, we construct several pairs of the adjacency, the Laplacian, the normalized Laplacian cospectral hypergraphs by using the tensor join
- …