377 research outputs found
Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields
A maximal minor of the Laplacian of an -vertex Eulerian digraph
gives rise to a finite group
known as the sandpile (or critical) group of . We determine
of the generalized de Bruijn graphs with
vertices and arcs for and , and closely related generalized Kautz graphs, extending and
completing earlier results for the classical de Bruijn and Kautz graphs.
Moreover, for a prime and an -cycle permutation matrix
we show that is isomorphic to the
quotient by of the centralizer of in
. This offers an explanation for the coincidence of
numerical data in sequences A027362 and A003473 of the OEIS, and allows one to
speculate upon a possibility to construct normal bases in the finite field
from spanning trees in .Comment: I+24 page
Learning Local Receptive Fields and their Weight Sharing Scheme on Graphs
We propose a simple and generic layer formulation that extends the properties
of convolutional layers to any domain that can be described by a graph. Namely,
we use the support of its adjacency matrix to design learnable weight sharing
filters able to exploit the underlying structure of signals in the same fashion
as for images. The proposed formulation makes it possible to learn the weights
of the filter as well as a scheme that controls how they are shared across the
graph. We perform validation experiments with image datasets and show that
these filters offer performances comparable with convolutional ones.Comment: To appear in 2017, 5th IEEE Global Conference on Signal and
Information Processing, 5 pages, 3 figures, 3 table
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
A digraph such that every proper induced subdigraph has a kernel is said to
be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI
for short) resp.) if the digraph has a kernel (does not have a kernel resp.).
The unique CKI-tournament is and the unique
KP-tournaments are the transitive tournaments, however bipartite tournaments
are KP. In this paper we characterize the CKI- and KP-digraphs for the
following families of digraphs: locally in-/out-semicomplete, asymmetric
arc-locally in-/out-semicomplete, asymmetric -quasi-transitive and
asymmetric -anti-quasi-transitive -free and we state that the problem
of determining whether a digraph of one of these families is CKI is polynomial,
giving a solution to a problem closely related to the following conjecture
posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for
locally in-semicomplete digraphs.Comment: 13 pages and 5 figure
An algebraic analysis of the graph modularity
One of the most relevant tasks in network analysis is the detection of
community structures, or clustering. Most popular techniques for community
detection are based on the maximization of a quality function called
modularity, which in turn is based upon particular quadratic forms associated
to a real symmetric modularity matrix , defined in terms of the adjacency
matrix and a rank one null model matrix. That matrix could be posed inside the
set of relevant matrices involved in graph theory, alongside adjacency,
incidence and Laplacian matrices. This is the reason we propose a graph
analysis based on the algebraic and spectral properties of such matrix. In
particular, we propose a nodal domain theorem for the eigenvectors of ; we
point out several relations occurring between graph's communities and
nonnegative eigenvalues of ; and we derive a Cheeger-type inequality for the
graph optimal modularity
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