Let $G$ be a connected graph on $n$ vertices and $1 \le k \le n-1$ an integer. The $k$-token graph of $G$ is the graph $F_k(G)$ whose vertices are all the $k$-subsets of vertices of $G$, two of which are adjacent whenever their symmetric difference is an edge of $G$. Every automorphism of $G$ induces an automorphism of $F_k(G)$ in a natural way. Suppose that $S:=\{x,y\}$ is a cut set of $G$, such that $x$ and $y$ have the same neighbours in $G\setminus \{x,y\}$. In this paper we show that there exist a large number of automorphisms of $F_k(G)$ defined by $S$ that are not induced by automorphisms of $G$. We also describe the group produced by all such $2$-cuts of $G$.
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\star G_2$. Using results on automorphisms, we study the d…
In this paper, we study the graph isomorphism and graph automorphism problems. We propose a novel technique to analyze graph isomorphism and graph automorphism. Further we handled some strongly regular datasets for prove the efficiency of our technique. The neighbourhood matrix $ \mathcal{NM}(G) $ was proposed in \cite {ALPaper} as a novel representation of graphs and was defined using the neighbourhood sets of the vertices. It was also shown that the matrix exhibits a bijection between the product of two well known graph matrices, namely the adjacency matrix and the Laplacian matrix. Further,…
Let $X$ be a subgroup of the full automorphism group of the Hamming graph $H(m,q)$, and $C$ a subset of the vertices of the Hamming graph. We say that $C$ is an \emph{$(X,2)$-neighbour transitive code} if $X$ is transitive on $C$, as well as $C_1$ and $C_2$, the sets of vertices which are distance $1$ and $2$ from the code. This paper begins the classification of $(X,2)$-neighbour transitive codes where the action of $X$ on the entries of the Hamming graph has a non-trivial kernel. There exists a subgroup of $X$ with a $2$-transitive action on the alphabet; this action is thus almost-simple or…
This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.
Let $X$ be a connected Cayley graph on an abelian group of odd order, such that no two distinct vertices of $X$ have exactly the same neighbours. We show that the direct product $X \times K_2$ (also called the "canonical double cover" of $X$) has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors $X$ and $K_2$). This means that $X$ is "stable". The proof is short and elementary. The theory of direct products implies that $K_2$ can be replaced with members of a much more general family of connected graphs.
A graph $X$ is said to be unstable if the direct product $X\times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is non-trivially unstable if it is unstable, connected, non-bipartite, and distinct vertices have distinct sets of neighbours. In this paper, we prove two sufficient conditions for stability of graphs in which every edge lies on a triangle, revising an incorrect claim of Surowski and filling in some gaps in the proof of another one. We also consider triangle-free graphs, and prove that there…
We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(Γ,Σ)$ is stable if $Aut(Γ\timesΣ) \cong Aut(Γ)\times Aut(Σ)$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. An unstable graph pair $(Γ,Σ)$ is said to be a nontrivially unstable graph pair if $Γ$ and $Σ$ are connected coprime graphs, at least one of them is non-bipartite, and each of them has the property that different vertices have distinct neighbourhoods. We obtain necessary conditions for a pa…
A code is a subset of the vertex set of a Hamming graph. The set of $s$-neighbours of a code is the set of all vertices at Hamming distance $s$ from their nearest codeword. A code $C$ is $s$-elusive if there exists a distinct code $C'$ that is equivalent to $C$ under the full automorphism group of the Hamming graph such that $C$ and $C'$ have the same set of $s$-neighbours. It is proved here that the minimum distance of an $s$-elusive code is at most $2s+2$, and that an $s$-elusive code with minimum distance at least $2s+1$ gives rise to a $q$-ary $t$-design with certain parameters. This leads…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are the power graph, enhanced power graph, deep commuting graph, commuting graph, and non-generating graph, though I give a briefer discussion of the nilpotence and solvability graphs, and make some remarks on more general graphs. Aspects to be discussed include induced subgraphs, forbidden subgraphs, connectedness, and automorphism groups. We can also ask about…
A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular coverings in which this homomorphism is prescribed by an action of a semiregular subgroup $Γ$ of $\textrm{Aut}(G)$; so $H \cong G / Γ$. In this paper, we study the behaviour of regular graph covering with respect to 1-cuts and 2-cuts in $G$. We describe reductions which produce a series of graphs $G = G_0,\dots,G_r$ such that $G_{i+1}$ is created from $G_i$ by replacing certain inclusion minimal subgraphs with colored edges. The process ends with a primitive graph $G_r$ which i…
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics（甘肃省数学与统计基础学科研究中心）
