Two-fold automorphisms of graphs

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.

2018

In this supplement, we will assume that all graphs are undirected graphs with no loops or multiple edges. In graph theory, we talk about graph isomorphisms. As a reminder, an isomorphism between graphs G and H is a bijection φ : V (G) → V (H) such that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(H). A graph automorphism is simply an isomorphism from a graph to itself. In other words, an automorphism on a graph G is a bijection φ : V (G) → V (G) such that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G). This definition generalizes to digraphs, multigraphs, and graph with loops. Let Aut(G) denote the set of all automorphisms on a graph G. Note that this forms a group under function composition. In other words,

1999

In computational complexity theory, a function f is called b(n)-enumerable if there exists a polynomial-time function which can restrict the output of f (x) to one of b(n) possible values. This paper investigates #GA, the function which computes the number of automorphisms of an undirected graph, and GI, the set of pairs of isomorphic graphs. The results in this paper show the following connections between the enumerability of #GA and the computational complexity of GI.

In this paper, we deduce some properties of f -sets of connected graphs. Also, we introduce the concept of fixing share of each vertex of a fixing set D to see the participation of each vertex when fixing a connected graph G. We define a parameter, called the fixing percentage, by using the concept of fixing share, which is helpful in determining the measure of the amount of fixing done by the elements of D in G.

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.

European Journal of Combinatorics, 2013

Let G g,b be the set of all uni/trivalent graphs representing the combinatorial structures of pant decompositions of the oriented surface Σ g,b of genus g with b boundary components. We describe the set A g,b of all automorphisms of graphs in G g,b showing that, up to suitable moves changing the graph within G g,b , any such automorphism can be reduced to elementary switches of adjacent edges.

arXiv: Combinatorics, 2020

In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information concerning these graphs that would be useful, as well as making explicit the extensions of these results to digraphs. Additionally, there are several small errors in some of the papers that were involved in this classification. The purpose of this paper is to fill in the missing information as well as correct all known errors.

On counting of labeled connected graphs, the question comes into mind first; "How many ways a graph can be labeled?" As there exist certain number of labeled isomorphic graphs. On finding non isomorphic labeled graphs is an interesting problem itself. To provide the fact, graph automorphism should be considered.

Facta Universitatis, Series: Mathematics and Informatics, 2021

The power graph of a group $G$ is the graph with vertex set $G$,having an edge joining $x$ and $y$ whenever one is a power of theother. The purpose of this paper is to study the automorphismgroups of the power graphs of infinite groups.

Graphs and Combinatorics, 1994

In this paper we review the characterization of point-color symmetric (PCS) graphs based on the color preserving automorphisms given in [3"1. In particular, we consider PCS pictures, arriving at another characterization theorem. We summarize a few results and give some examples.