Automorphism Group Of Graph Research Articles

Group-invariant Percolation on Graphs

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is developed further here.¶ Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group.¶ We show that G is amenable if for all $ \alpha < 1 $ , there is a G-invariant site percolation process $ \omega $ on X with $ {\bf P} [x \in \omega] > \alpha $ for all vertices x and with no infinite components. When G is not amenable, a threshold $ \alpha < 1 $ appears. An inequality for the threshold in terms of the isoperimetric constant is obtained, extending an inequality of Häggström for regular trees.¶ If G acts transitively on X, we show that G is unimodular if the expected degree is at least 2 in any G-invariant bond percolation on X with all components infinite.¶ The investigation of dependent percolation also yields some results on automorphism groups of graphs that do not involve percolation.

Automorphism Groups of Graphs with Quadratic Growth

LetΓbe a graph with almost transitive group Aut(Γ) and quadratic growth. We show that Aut(Γ) contains an almost transitive subgroup isomorphic to the free abelian group Z2.

Representing Groups by Colourings of Graphs

The topic of this paper is representing groups by edge-coloured graphs. Every edge-coloured graph determines a group of graph automorphisms which preserve the colours of the edges. An edge colouring of a graph G is called a perfect one iff every colour class is a perfect matching in G. We prove that every group H and all of its subgroups can be represented (up to isomorphism) by a group of colour preserving automorphisms related to some perfect colouring of the same graph.

Automorphism groups of graphs with 1-factorizations

A 1-factorization ϕ of a simple undirected connected graph G is an edge colouring such that each vertex is incident with exactly one edge of each colour. The automorphisms which preserve the colours of all edges constitute a group A c ( G, ϕ). We prove every finitely generated group H to be isomorphic to the full group A c( G, ϕ) for a regular graph G of degree 3 with a 1-factorization ϕ. Moreover we show that for every finitely generated group H there is a regular graph G of degree 5 such that the group H and all of its subgroups can be represented (up to isomorphism) by a group of colour preserving automorphisms related to some 1-factorization ϕ of G.

Some topological properties of star connected cycles

Star connected cycles are shown to be an undirected Cayley graph, and the graph automorphism group is determined. A routing algorithm is given, which finds an optimal path in polynomial time. The diameter and tight bounds on the average distance are computed.

Enumeration of graph embeddings

For a finite connected simple graph G, let Γ be a group of graph automorphisms of G. Two 2-cell embeddings ι: G → S and j: G → S of a graph G into a closed surface S (orientable or nonorientable) are congruent with respect to Γ if there are a surface homeomorphism h: S → S and a graph automorphism γϵΓ such that hoι= joγ. In this paper, we give an algebraic characterization of congruent 2-cell embeddings, from which we enumerate the congruence classes of 2-cell embeddings of a graph G into closed surfaces with respect to a group of automorphisms of G, not just the full automorphism group. Some applications to complete graphs are also discussed. As an orientable case, the oriented congruence of a graph G into orientable surfaces with respect to the full automorphism group of G was enumerated by Mull et al. (1988).

Computational Techniques for the Automorphism Groups of Graphs

Computational Techniques for the Automorphism Groups of Graphs

Symmetry of chemical structures: A novel method of graph automorphism group determination

Symmetry of chemical structures: A novel method of graph automorphism group determination

Automorphism groups of graphs with forbidden subgraphs

For a graph ? letF(?) be the class of finite graphs which do not contain an induced subgraph isomorphic to ?. We show that whenever ? is not isomorphic to a path on at most 4 vertices or to the complement of such a graph then for every finite groupG there exists a graph ??F(?) such thatG is isomorphic to the automorphism group of ?. For all paths ? on at most 4 vertices we determine the class of all automorphism groups of members ofF(?).

A survey on graphs with polynomial growth

In this paper we give an overview on connected locally finite transitive graphs with polynomial growth. We present results concerning the following topics: •Automorphism groups of graphs with polyn...

A survey on graphs with polynomial growth

In this paper we give an overview on connected locally finite transitive graphs with polynomial growth. We present results concerning the following topics: •Automorphism groups of graphs with polynomial growth.•Groups and graphs with linear growth.•S-transitivity.•Covering graphs.•Automorphism groups as topological groups.

Automorphism groups of graphs as topological groups

Automorphism groups of graphs as topological groups

Automorphism groups of graphs

Automorphism groups of graphs

Spectral analysis of graphs by cyclic automorphism subgroups

The theory of spectral decomposition modulo subgroups of the graph automorphism group is extended to cyclic configurations of arbitrary rotational order. By regarding graphs with cyclic automorphisms as composite polymers of relatively simple monomeric structural units, it is shown that the spectrum of eigenvalues of many prominent molecular and nonmolecular families devolves to consideration of a single monomer-derived reduction network. As the only parameter associated with this network is the set of simple circuit eigenvalues, a direct connection is forged between the spectrum of a circuit and the spectrum of any cyclic array of the same periodicity.

Some applications of graph contractions

AbstractResults in diverse areas, such as the Nielsen‐Schreier theorem on subgroups of free groups and a proof of A. T. White's conjecture on the genus of subgroups are shown to be immediate consequences of a lemma which has already proved useful in investigating topological properties and automorphism group of graphs.

Automorphism groups of graphs and edge-contraction

If a class C of finite graphs is closed under contraction and forming subgraphs, and if every finite abstract group occurs as the automorphism group of some graph in C, then C contains all finite graphs (up to isomorphism). Also related results concerning automorphism groups of graphs on given surfaces are mentioned.

On the action of non-Abelian groups on graphs

Finite groups may be classified as to whether or not they have regular representations as automorphism groups of graphs. Such a classification is begun here for non-Abelian groups (classification of Abelian groups being already in the literature) with a classification of the dihedral groups, the groups of order p 3 for p an odd prime (except for one group of order 27), and the generalized dicyclic groups. It is further shown that the class of groups (other than C 2) having the above regular representations is closed under the operation of direct product.

The group of an X-join of graphs

In this paper we give necessary and sufficient conditions for the group of graph automorphisms of the X-join of { Y x} x∈X to be the “natural” ones, i.e., those that are obtained by first permuting the Y x according to a permutation of subscripts by an automorphism of X and then performing an arbitrary automorphism of each Y x. This simultaneously solves and generalizes the problem of giving necessary and sufficient conditions for the group of the lexicographic product of two graphs to be the wreath product of the groups of the factor graphs. This problem was introduced by Harary and partial solutions had been given by Sabidussi and the author.

Dissimilarity characteristic theorems for graphs

A simple but general dissimilarity characteristic theorem was used in [i] to derive gerLerating functions for counting various classes of graphs. We propose to use this result in a slightly generalized form, stated below as Theorem, to give new and short derivations of Otter's dissimilarity theorem [7], its generalization to cacti [3] (formerly called Husimi trees), and an elementary theorem for graphs [2]. In general we follow standard terminology on graphs (K6nig [6]); our graphs are finite, nonempty, and may have isolated points. A block of a graph (Glied in [6]) is a maximal connected subgraph containing no cut of itself. For a given subgroup ?f of automorphism group of graph, we say that two are similar if a permutation of ?f sends one point into other. By the number of dissimilar points of a graph we mean number of equivalence classes of similar in it; analogously for lines, blocks, etc.