7,590 research outputs found
On Distance-Regular Graphs with Smallest Eigenvalue at Least
A non-complete geometric distance-regular graph is the point graph of a
partial geometry in which the set of lines is a set of Delsarte cliques. In
this paper, we prove that for fixed integer , there are only finitely
many non-geometric distance-regular graphs with smallest eigenvalue at least
, diameter at least three and intersection number
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
A bilinear form relating two Leonard systems
Let , be Leonard systems over a field , and ,
the vector spaces underlying , , respectively. In this paper,
we introduce and discuss a balanced bilinear form on . Such a form
naturally arises in the study of -polynomial distance-regular graphs. We
characterize a balanced bilinear form from several points of view.Comment: 15 page
Tensor invariants for certain subgroups of the orthogonal group
Let V be an n-dimensional vector space and let On be the orthogonal group.
Motivated by a question of B. Szegedy (B. Szegedy, Edge coloring models and
reflection positivity, Journal of the American Mathematical Society Volume 20,
Number 4, 2007), about the rank of edge connection matrices of partition
functions of vertex models, we give a combinatorial parameterization of tensors
in V \otimes k invariant under certain subgroups of the orthogonal group. This
allows us to give an answer to this question for vertex models with values in
an algebraically closed field of characteristic zero.Comment: 14 pages, figure. We fixed a few typo's. To appear in Journal of
Algebraic Combinatoric
Expanders and right-angled Artin groups
The purpose of this article is to give a characterization of families of
expander graphs via right-angled Artin groups. We prove that a sequence of
simplicial graphs forms a family of expander
graphs if and only if a certain natural mini-max invariant arising from the cup
product in the cohomology rings of the groups
agrees with the Cheeger constant of the
sequence of graphs, thus allowing us to characterize expander graphs via
cohomology. This result is proved in the more general framework of \emph{vector
space expanders}, a novel structure consisting of sequences of vector spaces
equipped with vector-space-valued bilinear pairings which satisfy a certain
mini-max condition. These objects can be considered to be analogues of expander
graphs in the realm of linear algebra, with a dictionary being given by the cup
product in cohomology, and in this context represent a different approach to
expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.Comment: 21 pages. Accepted version. To appear in J. Topol. Ana
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