2,690 research outputs found
Jordan-like characterization of automorphism groups of planar graphs
We investigate automorphism groups of planar graphs. The main result is a
complete recursive description of all abstract groups that can be realized as
automorphism groups of planar graphs. The characterization is formulated in
terms of inhomogeneous wreath products. In the proof, we combine techniques
from combinatorics, group theory, and geometry. This significantly improves the
Babai's description (1975).Comment: Final version of the paper. A lot of conceptual changes were made in
comparison to the previous version
On quartic half-arc-transitive metacirculants
Following Alspach and Parsons, a {\em metacirculant graph} is a graph
admitting a transitive group generated by two automorphisms and
, where is -semiregular for some integers , , and where normalizes , cyclically permuting the orbits
of in such a way that has at least one fixed vertex. A {\em
half-arc-transitive graph} is a vertex- and edge- but not arc-transitive graph.
In this article quartic half-arc-transitive metacirculants are explored and
their connection to the so called tightly attached quartic half-arc-transitive
graphs is explored. It is shown that there are three essentially different
possibilities for a quartic half-arc-transitive metacirculant which is not
tightly attached to exist. These graphs are extensively studied and some
infinite families of such graphs are constructed.Comment: 31 pages, 2 figure
Cubic symmetric graphs having an abelian automorphism group with two orbits
Finite connected cubic symmetric graphs of girth 6 have been classified by K.
Kutnar and D. Maru\v{s}i\v{c}, in particular, each of these graphs has an
abelian automorphism group with two orbits on the vertex set. In this paper all
cubic symmetric graphs with the latter property are determined. In particular,
with the exception of the graphs K_4, K_{3,3}, Q_3, GP(5,2), GP(10,2), F40 and
GP(24,5), all the obtained graphs are of girth 6.Comment: Keywords: cubic symmetric graph, Haar graph, voltage graph. This
papaer has been withdrawn by the author because it is an outdated versio
Symmetry Type Graphs of Abstract Polytopes and Maniplexes
A -orbit maniplex is one that has orbits of flags under the action of
its automorphism group. In this paper we extend the notion of symmetry type
graphs of maps to that of maniplexes and polytopes and make use of them to
study -orbit maniplexes, as well as fully-transitive 3-maniplexes. In
particular, we show that there are no fully-transtive -orbit 3-mainplexes
with an odd number, we classify 3-orbit mainplexes and determine all
face transitivities for 3- and 4-orbit maniplexes. Moreover, we give generators
of the automorphism group of a polytope or a maniplex, given its symmetry type
graph. Finally, we extend these notions to oriented polytopes, in particular we
classify oriented 2-orbit maniplexes and give generators for their orientation
preserving automorphism group
Cubic vertex-transitive graphs on up to 1280 vertices
A graph is called cubic and tetravalent if all of its vertices have valency 3
and 4, respectively. It is called vertex-transitive and arc-transitive if its
automorphism group acts transitively on its vertex-set and on its arc- set,
respectively. In this paper, we combine some new theoretical results with
computer calculations to construct all cubic vertex-transitive graphs of order
at most 1280. In the process, we also construct all tetravalent arc-transitive
graphs of order at most 640
A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two
A complete list of all connected arc-transitive asymmetric digraphs of
in-valence and out-valence 2 on up to 1000 vertices is presented. As a
byproduct, a complete list of all connected 4-valent graphs admitting a
half-arc-transitive group of automorphisms on up to 1000 vertices is obtained.
Several graph-theoretical properties of the elements of our census are
calculated and discussed
Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. II. Program Implementation of the Weisfeiler-Leman Algorithm
The stabilization algorithm of Weisfeiler and Leman has as an input any
square matrix A of order n and returns the minimal cellular (coherent) algebra
W(A) which includes A.
In case when A=A(G) is the adjacency matrix of a graph G the algorithm
examines all configurations in G having three vertices and, according to this
information, partitions vertices and ordered pairs of vertices into equivalence
classes. The resulting construction allows to associate to each graph G a
matrix algebra W(G):= W(A(G))$ which is an invariant of the graph G. For many
classes of graphs, in particular for most of the molecular graphs, the algebra
W(G) coincides with the centralizer algebra of the automorphism group aut(G).
In such a case the partition returned by the stabilization algorithm is equal
to the partition into orbits of aut(G).
We give algebraic and combinatorial descriptions of the Weisfeiler--Leman
algorithm and present an efficient computer implementation of the algorithm
written in C. The results obtained by testing the program on a considerable
number of examples of graphs, in particular on some chemical molecular graphs,
are also included.Comment: Arxiv version of a preprint published in 199
Equitable Decompositions of Graphs
We investigate connections between the symmetries (automorphisms) of a graph
and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial
automorphism , it is possible to use to decompose any matrix
appropriately associated with the graph. The
result of this decomposition is a number of strictly smaller matrices whose
collective eigenvalues are the same as the eigenvalues of the original matrix
. Some of the matrices that can be decomposed are the graph's adjaceny
matrix, Laplacian matrix, etc. Because this decomposition has connections to
the theory of equitable partitions it is referred to as an equitable
decomposition. Since the graph structure of many real-world networks is quite
large and has a high degree of symmetry, we discuss how equitable
decompositions can be used to effectively bound both the network's spectral
radius and spectral gap, which are associated with dynamic processes on the
network. Moreover, we show that the techniques used to equitably decompose a
graph can be used to bound the number of simple eigenvalues of undirected
graphs, where we obtain sharp results of Petersdorf-Sachs type
Vsep-New Heuristic and Exact Algorithms for Graph Automorphism Group Computation
One exact and two heuristic algorithms for determining the generators, orbits
and order of the graph automorphism group are presented. A basic tool of these
algorithms is the well-known individualization and refinement procedure. A
search tree is used in the algorithms - each node of the tree is a partition.
All nonequivalent discreet partitions derivative of the selected vertices are
stored in a coded form. A new strategy is used in the exact algorithm: if
during its execution some of the searched or intermediate variables obtain a
wrong value then the algorithm continues from a new start point losing some of
the results determined so far. The algorithms has been tested on one of the
known benchmark graphs and shows lower running times for some graph families.
The heuristic versions of the algorithms are based on determining some number
of discreet partitions derivative of each vertex in the selected cell of the
initial partition and comparing them for an automorphism - their search trees
are reduced. The heuristic algorithms are almost exact and are many times
faster than the exact one. The experimental tests exhibit that the worst-cases
running time of the exact algorithm is exponential but it is polynomial for the
heuristic algorithms. Several cell selectors are used. Some of them are new. We
also use a chooser of cell selector for choosing the optimal cell selector for
the manipulated graph. The proposed heuristic algorithms use two main heuristic
procedures that generate two different forests of search trees.Comment: 47 pages; 1. Entirely revised 2. Algorithms analysis removed 3. New
algorithm versions added, one version removed 4. Changed algorithm COMP -
cases CS2/CS4 are solved in a new wa
Finite edge-transitive oriented graphs of valency four: a global approach
We develop a new framework for analysing finite connected, oriented graphs of
valency 4, which admit a vertex-transitive and edge-transitive group of
automorphisms preserving the edge orientation. We identify a sub-family of
"basic" graphs such that each graph of this type is a normal cover of at least
one basic graph. The basic graphs either admit an edge-transitive group of
automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit
an (oriented or unoriented) cycle as a normal quotient. We anticipate that each
of these additional properties will facilitate effective further analysis, and
we demonstrate that this is so for the quasiprimitive basic graphs. Here we
obtain strong restirictions on the group involved, and construct several
infinite families of such graphs which, to our knowledge, are different from
any recorded in the literature so far. Several open problems are posed in the
paper.Comment: 19 page
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