Automorphism groups of graph covers and uniform subset graphs

Doklady Mathematics, 2008

We consider nonoriented graphs without loops and multiple edges. For a vertex a of a graph Γ , by Γ i (a) we denote the i-neighborhood of a , i.e., the subgraph induced by Γ on the set of all vertices at distance i from a. We set [ a ] = Γ 1 (a) and a ⊥ = { a } ∪ [ a ]. For a graph Γ and a , b ∈ Γ , we denote the number of vertices in [ a ] ∩ [ b ] by µ (a , b) (by λ (a , b)) if a and b are distance 2 apart (adjacent) in Γ. If Γ is a graph of diameter d and I ⊂ {1, 2, …, d } , then Γ I denotes the graph on the same vertices as Γ in which two vertices u and w are adjacent if and only if d (u , w) ∈ I .

1998

Abstract A distance-transitive antipodal cover of a complete graph Kn possesses an automorphism group that acts 2-transitively on the fibres. The classification of finite simple groups implies a classification of finite 2-transitive permutation groups, and this allows us to determine all possibilities for such a graph. Several new infinite families of distance transitive graphs are constructed.

Doklady Mathematics, 2008

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).

Doklady Mathematics, 2010

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.

Journal of Algebraic Combinatorics, 2014

We give a general construction leading to different non-isomorphic families Γ n,q (K) of connected q-regular semisymmetric graphs of order 2q n+1 embedded in PG(n + 1, q), for a prime power q = p h , using the linear representation of a particular point set K of size q contained in a hyperplane of PG(n + 1, q). We show that, when K is a normal rational curve with one point removed, the graphs Γ n,q (K) are isomorphic to the graphs constructed for q = p h in Lazebnik and Viglione (J. Graph Theory 41, [249][250][251][252][253][254][255][256][257][258] 2002) and to the graphs constructed for q prime in Du et al. (Eur. J. Comb. 24, 897-902, 2003). These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For q ≥ n + 3 or q = p = n + 2, n ≥ 2, we obtain their full automorphism group from our construction by showing that, for an arc K, every automorphism of Γ n,q (K) is induced by a collineation of the ambient space PG(n + 1, q). We also give some other examples of semisymmetric graphs Γ n,q (K) for which not every automorphism is induced by a collineation of their ambient space.

Discrete Mathematics, 2004

A directed graph is k-arc transitive if it has an automorphism group which acts transitively on the set of k-arcs. Several techniques have been proposed in the literature to construct k-arc transitive digraphs for each positive integer k and each degree r of regularity. Following previous work by the authors, we study the full automorphism group of k-arc transitive digraphs and we give its structure for digraphs of degree 2. We describe the k-transitive coverings of complete digraphs which can be obtained by the construction technique introduced by the authors in previous work. Finally, we provide examples of k-arc transitive digraphs which are non-homomorphic to a cycle.

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.

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.