79,186 research outputs found
Classification of quasi-symmetric 2-(64,24,46) designs of Blokhuis-Haemers type
This paper completes the classification of quasi-symmetric 2-
designs of Blokhuis-Haemers type supported by the dual code of the
binary linear code spanned by the lines of initiated in
\cite{bgr-vdt}. It is shown that contains exactly 30,264
nonisomorphic quasi-symmetric 2- designs obtainable from maximal
arcs in via the Blokhuis-Haemers construction. The related strongly
regular graphs are also discussed.Comment: 11 page
Exploring Structure-Adaptive Graph Learning for Robust Semi-Supervised Classification
Graph Convolutional Neural Networks (GCNNs) are generalizations of CNNs to
graph-structured data, in which convolution is guided by the graph topology. In
many cases where graphs are unavailable, existing methods manually construct
graphs or learn task-driven adaptive graphs. In this paper, we propose Graph
Learning Neural Networks (GLNNs), which exploit the optimization of graphs (the
adjacency matrix in particular) from both data and tasks. Leveraging on
spectral graph theory, we propose the objective of graph learning from a
sparsity constraint, properties of a valid adjacency matrix as well as a graph
Laplacian regularizer via maximum a posteriori estimation. The optimization
objective is then integrated into the loss function of the GCNN, which adapts
the graph topology to not only labels of a specific task but also the input
data. Experimental results show that our proposed GLNN outperforms
state-of-the-art approaches over widely adopted social network datasets and
citation network datasets for semi-supervised classification
Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs
A finite simple graph is called a bi-Cayley graph over a group if it has
a semiregular automorphism group, isomorphic to which has two orbits on
the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups
have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014),
679--693). In this paper we consider the latter class of graphs and select
those in the class which are also arc-transitive. Furthermore, such a graph is
called -type when it is bipartite, and the bipartition classes are equal to
the two orbits of the respective semiregular automorphism group. A -type
graph can be represented as the graph where is a
subset of the vertex set of which consists of two copies of say
and and the edge set is . A
bi-Cayley graph is called a BCI-graph if for any bi-Cayley
graph
implies that for some and . It is also shown that every cubic connected arc-transitive
-type bi-Cayley graph over an abelian group is a BCI-graph
On the orders of arc-transitive graphs
A graph is called {\em arc-transitive} (or {\em symmetric}) if its
automorphism group has a single orbit on ordered pairs of adjacent vertices,
and 2-arc-transitive its automorphism group has a single orbit on ordered paths
of length 2. In this paper we consider the orders of such graphs, for given
valency. We prove that for any given positive integer , there exist only
finitely many connected 3-valent 2-arc-transitive graphs whose order is
for some prime , and that if , then there exist only finitely many
connected -valent 2-arc-transitive graphs whose order is or for
some prime . We also prove that there are infinitely many (even) values of
for which there are only finitely many connected 3-valent symmetric graphs
of order where is prime
The Morita theory of quantum graph isomorphisms
We classify instances of quantum pseudo-telepathy in the graph isomorphism
game, exploiting the recently discovered connection between quantum information
and the theory of quantum automorphism groups. Specifically, we show that
graphs quantum isomorphic to a given graph are in bijective correspondence with
Morita equivalence classes of certain Frobenius algebras in the category of
finite-dimensional representations of the quantum automorphism algebra of that
graph. We show that such a Frobenius algebra may be constructed from a central
type subgroup of the classical automorphism group, whose action on the graph
has coisotropic vertex stabilisers. In particular, if the original graph has no
quantum symmetries, quantum isomorphic graphs are classified by such subgroups.
We show that all quantum isomorphic graph pairs corresponding to a well-known
family of binary constraint systems arise from this group-theoretical
construction. We use our classification to show that, of the small order
vertex-transitive graphs with no quantum symmetry, none is quantum isomorphic
to a non-isomorphic graph. We show that this is in fact asymptotically almost
surely true of all graphs.Comment: 53 pages. V2: changed drawing conventions to avoid usage of opposite
algebra
On basic graphs of symmetric graphs of valency five
A graph \G is {\em symmetric} or {\em arc-transitive} if its automorphism
group \Aut(\G) is transitive on the arc set of the graph, and \G is {\em
basic} if \Aut(\G) has no non-trivial normal subgroup such that the
quotient graph \G_N has the same valency with \G. In this paper, we
classify symmetric basic graphs of order and valency 5, where are
two primes and is a positive integer. It is shown that such a graph is
isomorphic to a family of Cayley graphs on dihedral groups of order with
5\di (q-1), the complete graph of order , the complete bipartite
graph of order 10, or one of the nine sporadic coset graphs
associated with non-abelian simple groups. As an application, connected
pentavalent symmetric graphs of order for some small integers and
are classified
On cubic symmetric non-Cayley graphs with solvable automorphism groups
It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs
with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015),
1-11] that a cubic symmetric graph with a solvable automorphism group is either
a Cayley graph or a -regular graph of type , that is, a graph with no
automorphism of order interchanging two adjacent vertices. In this paper an
infinite family of non-Cayley cubic -regular graphs of type with a
solvable automorphism group is constructed. The smallest graph in this family
has order 6174.Comment: 8 page
Nilpotent dessins: Decomposition theorem and classification of the abelian dessins
A map is a 2-cell decomposition of an orientable closed surface. A dessin is
a bipartite map with a fixed colouring of vertices. A dessin is regular if its
group of colour- and orientation-preserving automorphisms acts transitively on
the edges, and a regular dessin is symmetric if it admits an additional
external symmetry transposing the vertex colours. Regular dessins with
nilpotent automorphism groups are investigated. We show that each such dessin
is a parallel product of regular dessins whose automorphism groups are the
Sylow subgroups. Regular and symmetric dessins with abelian automorphism groups
are classified and enumerated.Comment: 27page
Enumeration of Seidel matrices
In this paper Seidel matrices are studied, and their spectrum and several
related algebraic properties are determined for order . Based on this
Seidel matrices with exactly three distinct eigenvalues of order are
classified. One consequence of the computational results is that the maximum
number of equiangular lines in with common angle is
exactly .Comment: 18 pages, 9 table
Automorphism groups of circulant graphs -- a survey
A circulant (di)graph is a (di)graph on n vertices that admits a cyclic
automorphism of order n. This paper provides a survey of the work that has been
done on finding the automorphism groups of circulant (di)graphs, including the
generalisation in which the edges of the (di)graph have been assigned colours
that are invariant under the aforementioned cyclic automorphism.Comment: 16 pages, 0 figures, LaTeX fil
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