Automorphisms of trivalent graphs

In this paper we shall present a natural generalisation of the notion of automorphism of a graph or digraph G, namely a two-fold automorphism. This is a pair (α, β) of permutations of the vertex set V(G) which acts on ordered pairs of vertices of G in the natural way. The action of (α, β) on all such ordered pairs gives a graph which is two-fold isomorphic to G. When α = β the two-fold automorphism is just a usual automor-phism. Our main results concern those graphs which admit a two-fold automorphism with α = β.

2022

In this paper, we study the relationship between the mapping class group of an infinite-type surface and the simultaneous flip graph, a variant of the flip graph for infinite-type surfaces defined by Fossas and Parlier [8]. We show that the extended mapping class group is isomorphic to a proper subgroup of the automorphism group of the flip graph, unlike in the finite-type case. This shows that Ivanov’s metaconjecture, which states that any “sufficiently rich” object associated to a finite-type surface has the extended mapping class group as its automorphism group, does not extend to simultaneous flip graphs of infinite-type surfaces.

Acta Mathematica Sinica-english Series, 2005

A graph is called a semi–regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi–regular. In this paper, a necessary and sufficient condition for an automorphism of the graph Γ to be an automorphism of a map with the underlying graph Γ is obtained. Using this result, all orientation–preserving automorphisms of maps on surfaces (orientable and non–orientable) or just orientable surfaces with a given underlying semi–regular graph Γ are determined. Formulas for the numbers of non–equivalent embeddings of this kind of graphs on surfaces (orientable, non–orientable or both) are established, and especially, the non–equivalent embeddings of circulant graphs of a prime order on orientable, non–orientable and general surfaces are enumerated.

This paper introduces the basic definitions and properties of simple graphs which are mainly covered in [1] and [2]. Each definition and property is supported by examples and diagrams. There are also some basic facts used in this paper which have been demonstrated by other researchers such as [3] and [4]. The main concern and the focus in this paper are on the automorphism groups of some graphs. The final part of this work have been on cubic graphs and the Boolian graph B n. To achieve the main points, the group automorphisms have been applied on the automorphisms of some graphs. The permutation groups played the principle role in the case. This was used to study the nature of the graph automorphisms.

Graphs and Combinatorics, 2006

A map is a connected topological graph cellularly embedded in a surface. For a given graph Γ, its genus distribution of rooted maps and embeddings on orientable and non-orientable surfaces are separately investigated by many researchers. By introducing the concept of a semi-arc automorphism group of a graph and classifying all its embeddings under the action of its semi-arc automorphism group, we find the relations between its genus distribution of rooted maps and genus distribution of embeddings on orientable and non-orientable surfaces, and give some new formulas for the number of rooted maps on a given orientable surface with underlying graph a bouquet of cycles B n , a closed-end ladder L n or a Ringel ladder R n . A general scheme for enumerating unrooted maps on surfaces(orientable or non-orientable) with a given underlying graph is established. Using this scheme, we obtained the closed formulas for the numbers of non-isomorphic maps on orientable or non-orientable surfaces with an underlying bouquet B n in this paper.

arXiv: Combinatorics, 2015

By Frucht's Theorem, every abstract finite group is isomorphic to the automorphism group of some graph. In 1975, Babai characterized which of these abstract groups can be realized as the automorphism groups of planar graphs. In this paper, we give a more detailed and understandable description of these groups. We describe stabilizers of vertices in connected planar graphs as the class of groups closed under the direct product and semidirect products with symmetric, dihedral and cyclic groups. The automorphism group of a connected planar graph is obtained as semidirect product of a direct product of these stabilizers with a spherical group. Our approach is based on the decomposition to 3-connected components and gives a quadratic-time algorithm for computing the automorphism group of a planar graph.

Discrete Mathematics, 1994

We identify three mutually nonisomorphic triangulations of the closed orientable surface of genus 20, each with the complete graph on 19 vertices.

2016

A modified Wiener number was proposed by Graovać and Pisanski. It is based on the full automorphism group of a graph. In this paper, we compute the difference between these topological indices for some polyhedral graphs.

2017

In 1975, Babai characterized which abstract groups can be realized as the automorphism groups of planar graphs. In this paper, we give a more detailed and understandable description of these groups. We describe stabilizers of vertices in connected planar graphs as the class of groups closed under the direct product and semidirect products with symmetric, dihedral and cyclic groups. The automorphism group of a connected planar graph is then obtained as a semidirect product of a direct product of these stabilizers with a spherical group. The formulation of the main result is new and original. Moreover, it gives a deeper in the structure of the groups. As a consequence, automorphism groups of several subclasses of planar graphs, including 2-connected planar, outerplanar, and series-parallel graphs, are characterized. Our approach translates into a quadratic-time algorithm for computing the automorphism group of a planar graph which is the first such algorithm described in detail.

Discrete Mathematics, 2006

Orientable triangular embeddings of the complete tripartite graph K n,n,n correspond to biembeddings of Latin squares. We show that if n is prime there are at least e n ln n−n(1+o(1)) nonisomorphic biembeddings of cyclic Latin squares of order n. If n = kp, where p is a large prime number, then the number of nonisomorphic biembeddings of cyclic Latin squares of order n is at least e p ln p−p(1+ln k+o(1)). Moreover, we prove that for every n there is a unique regular triangular embedding of K n,n,n in an orientable surface.