Related papers: On fixity of arc-transitive graphs
In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly…
We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph…
A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the…
A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let $\Gamma$ be a connected graph…
In this paper we consider connected locally $G$-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighourhood $\Gamma(u) \cup \Gamma(v)$ is trivial. We…
We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not…
It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a…
We consider a finite, connected and simple graph $\Gamma$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(\Gamma)$, the local action $G_x^{\Gamma(x)}$ is the action of…
Let $\Gamma$ be a connected $G$-arc-transitive graph, let $uv$ be an arc of $\Gamma$ and let $L$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. We study the problem of…
A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and…
This paper proposes a family of network centralities called fixed-point centralities. This centrality family is defined via the fixed point of permutation equivariant mappings related to the underlying network. Such a centrality notion is…
The stability number of a graph G is the cardinality of a stability system of G (that is of a stable set of maximum size of G). A graph is alpha-stable if its stability number remains the same upon both the deletion and the addition of any…
A graph or hypergraph is said to be vertex-transitive if its automorphism group acts transitively upon its vertices. A classic theorem of Mader asserts that every connected vertex-transitive graph is maximally edge-connected. We generalise…
An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…
The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. Firstly, the connective…
A non-complete graph $\Gamma$ is said to be $(G,2)$-distance transitive if $G$ is a subgroup of the automorphism group of $\Gamma$ that is transitive on the vertex set of $\Gamma$, and for any vertex $u$ of $\Gamma$, the stabilizer $G_u$ is…
A graph is called {\em half-arc-transitive} if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime $p$ there is no tetravalent half-arc-transitive graph of order $p$ or…
Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$, which is the maximum number of fixed points of an involution. In this paper we study the involution fixity of almost simple…
A walk of length $n$ in a graph is consistent if there exists an automorphism of the graph that maps the initial $n-1$ vertices to the final $n-1$ vertices of the walk. In this paper we find some sufficient conditions for a consistent walk…
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…