Two-fold orbital digraphs and other constructions

Let G be a finite non-abelian group and let Ω be a set of elements of G. Let A be the set of commuting elements in , i.e A = {v ∈ Ω : vg = gv, g ∈ G}. In this paper, we extend the work on conjugate graph by defining a new graph called the orbit graph, denoted as Γ^{Ω}_G. The vertices of Γ^{Ω}_G are non- central elements in Ω but not in A in which two vertices of Γ^{Ω}_G are adjacent if they are conjugate. Some graph properties are provided. Besides, the orbit graph of dihedral groups and quaternion groups is determined.

2009

Abstract We prove that, for a positive integer n and subgroup H of automorphisms of a cyclic group Z of order n, there is up to isomorphism a unique connected circulant digraph based on Z admitting an arc-transitive action of Z⋊ H. We refine the Kovács–Li classification of arc-transitive circulants to determine those digraphs with automorphism group larger than Z⋊ H.

Arxiv preprint math/0611758, 2006

If G is a group acting on a set V and a, b are elements of V, the digraph whose vertex set is V and whose arc set is the orbit (a, b)^G is called an orbital digraph of G. A locally finite digraph X has more than one end if there exists a finite set of vertices Y such that the induced digraph X \ Y contains at least two infinite connected components; if there exists such a set containing precisely one element, then X has connectivity one. In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivity-one orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterised in another paper (Infinite primitive digraphs) by the author.

Engineering Mathematics, 2024

The combinatorial properties and invariants which includes transitivity, primitivity, ranks and subdegrees of direct product of Alternating group and Cyclic group acting on Cartesian product of two set have been extensively studied. However, the construction of suborbital graphs for this group action remains largely unexplored. As a result, this research paper addresses this gap by constructing suborbital graphs involving direct product of the Alternating group and Cyclic group acting on the Cartesian product of two sets where their respective properties such as connectivity, self-pairedness, girth and vertex degree are analyzed in detail. First, the result shows that all suborbits are self − paired implies that the vertex sets are undirected to each other. Secondly, the constructed suborbital graphs are classified into three parts for any value of n 3. In part A, it is proven that the constructed suborbital graphs Γ 1 , Γ 2 , Γ 3 ,..., Γ (n − 1) are undirected, regular of degree (n−1), disconnected with n−connected components and has a girth of 3. In part B, the constructed suborbital graph Γ (n−1)+1 is found to be undirected, regular of degree (n − 1), disconnected with n − connected components and has a girth of 3. Lastly in part C, the graphs Γ (n−1)+2 , Γ (n−1)+3 , Γ (n−1)+4 ,..., Γ (n−1)+n are found to be undirected, regular of degree (n − 1) 2 , disconnected with n 2 − connected components and has a girth of 3.

Combinatorica, 2014

We resolve two problems of [Cameron, Praeger, and Wormald -Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each 'building block' is a finite bipartite graph that is not a disjoint union of complete bipartite graphs. This was conjectured impossible in the above paper. We also describe the structure of 2-ended highly arc transitive digraphs in more generality, although complete characterization remains elusive.

Discrete Mathematics, 2000

A simple combinatorial construction is given which takes as its imput a regular graph of valency k such that the convex closure of two points at distance two is the complete bipartite graph K3;3 and whose output is a regular graph of valency 2k + 1. It is shown that the sequence of graphs obtained by starting with the graph with one point and no edges and applying the construction recursively is the family of bipartite dual polar space of type DSO + (2n; 2).

Applied Mathematics Letters, 2010

Let D = (V , A) be a digraph. Consider X and Y (not necessarily disjoint), nonempty subsets of vertices of D. We define a disimplex K (X, Y) of D to be the subdigraph whose vertex set is V (K (X, Y)) = X ∪ Y and in which an arc goes from every vertex of X to every vertex of Y (when X ∩ Y = ∅, loops are not considered). A disimplex K (X, Y) is called a diclique of D if K (X, Y) is not a proper subdigraph of any other disimplex of D. The diclique digraph (or diclique operator) − → k (D) of a digraph D is the digraph whose vertex set is the set of all dicliques of D and K (X, Y), K (X , Y) is an arc of − → k (D) if and only if Y ∩ X = ∅. We say that a digraph D is self-diclique if − → k (D) is isomorphic to D. In this paper we exhibit an infinite family of self-diclique circulant digraphs for which one of its members is an Eulerian orientation of the graph of the regular octahedron. This family is a natural generalization of the example given in Zelinka (2002) [5].

Journal of Graph Theory, 2015

We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets L, R of a group G, we define the two-sided group digraph − → 2S(G; L, R) to have vertex set G, and an arc from x to y if and only if y = −1 xr for some ∈ L and r ∈ R. In common with Cayley graphs and digraphs, two-sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which − → 2S(G; L, R) may be viewed as a simple graph of valency |L| • |R|, and we call such graphs two-sided group graphs. We also give sufficient conditions for two-sided group digraphs to be connected, vertex-transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.

Discrete Mathematics, 2000

A simple combinatorial construction is given which takes as its imput a regular graph of valency k such that the convex closure of two points at distance two is the complete bipartite graph K3;3 and whose output is a regular graph of valency 2k + 1. It is shown that the sequence of graphs obtained by starting with the graph with one point and no edges and applying the construction recursively is the family of bipartite dual polar space of type DSO + (2n; 2).

2008

ABSTRACT: This paper introduces three new types of combinatorial structures associated with group actions, namely symmetrical covers, symmetrical decompositions, and symmetrical factorisations of graphs. Thesestructures are related to and generalise various combinatorial objects, such as 2-designs, regular maps, near-polygonal graphs, and linear spaces. General theory is developed for each of these structures, pertinent examples and constructions are given, and a number of open research problems are posed.