This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of ``prime cycles'' at a vertex. The bounds are obtained by constructing circuits in the graph that resemble ``cactus trees'', and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence and underestimation of its spectral radius.
The bounds obtained are very good for the Cayley graph of surface groups of genus g>=2, with standard generators (these graphs are the 1-skeletons of tessellations of hyperbolic plane by 4g-gons, 4g per vertex). We have for example for g=2
0.662420<=|M|<=0.662816, and for g=3
0.552773<=|M|<=0.552792.
We study groups acting vertex-transitively on connected, trivalent graphs such that stabilizers of vertices are infinite. If the action is edge-transitive, we prove that the graph has to be a tree. We analyze the case where the action is not edge-transitive and fully classify the possible $2$-ended graphs. We draw connections to Willis' scale function and re-prove a result by Trofimov.
We observe that a ball of radius $1$ in the grandfather graph can not be realized as a ball of radius $1$ in a finite vertex transitive graph. We remark on when a ball in a finite vertex transitive graph appears as a ball in an infinite vertex transitive graph.
A graph is called cubic and tetravalent if all of its vertices have valency 3 and 4, respectively. It is called vertex-transitive and arc-transitive if its automorphism group acts transitively on its vertex-set and on its arc- set, respectively. In this paper, we combine some new theoretical results with computer calculations to construct all cubic vertex-transitive graphs of order at most 1280. In the process, we also construct all tetravalent arc-transitive graphs of order at most 640.
It is shown that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order $6p$ which is not genuinely imprimitive contains a Hamilton cycle.
It is shown that every connected vertex-transitive graph of order $4p$, where $p$ is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path.
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.
The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study for GI and DGI-groups, defined analogously to the concept of CI and DCI-groups.
Let $p$ be a prime and let $L$ be either the intransitive permutation group $C_p\times C_p$ of degree $2p$ or the transitive permutation group $C_p \wr C_2$ of degree $2p$. Let $Γ$ be a connected $G$-vertex-transitive and $G$-edge-transitive graph and let $v$ be a vertex of $Γ$. We show that if the permutation group induced by the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$ is isomorphic to $L$ then either $|V(Γ)|\geq p|G_v|\log_p\left(|G_v|/2\right)$, or $|V(Γ)|$ is bounded by a constant depending only on $p$, or $Γ$ is a very-well understood graph. This generalises a few recent resul…
In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph $G(10,2)$, occurring as the graph $40$ in the Foster census of connected symmetric trivalent graphs.
A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.
The main result of this paper is that, if $Γ$ is a finite connected $4$-valent arc-transitive graph, then either $Γ$ is part of a well-understood family of graphs, or every non-identity automorphism of $Γ$ fixes at most $1/3$ of the vertices. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs. Based on these results we propose a conjecture on the number of fixed points of non-identity automorphisms of vertex-transitive graphs of bounded valency.
We consider a finite, connected and simple graph $Γ$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(Γ)$, the local action $G_x^{Γ(x)}$ is the action of $\mathrm{Sym}(n)$ on ordered pairs, we show that the group $G_x^{[3]}$, the pointwise stabiliser of a ball of radius three around $x$, is trivial.
Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani（卢布尔雅那大学数学与物理学院）；Mathematics Institute, University of Warwick（华威大学数学研究所）；Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca（米兰比可卡大学数学与应用系）
A graph is said to be {\em vertex-transitive non-Cayley} if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order $12p$, where $p$ is a prime, is given. As a result, there are $11$ sporadic and one infinite family of such graphs, of which the sporadic ones occur when $p=5$, $7$ or $17$, and the infinite family exists if and only if $p\equiv1\ (\mod 4)$, and in this family there is a unique graph for a given order.
In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a $G$-vertex-transitive graph $Γ$. In the main result the group $G$ is quasiprimitive or biquasiprimitive on the vertices of $Γ$, and we obtain a genuine reduction to the case where $G$ is a nonabelian simple group. Using normal quotient techniques developed by the first author, the main theorem applies to general $G$-vertex-transitive graphs which are $G$-locally primitive (respectively, $G$-locally quasiprimitive), that is, the stabiliser $G_α$ of a vertex $α$ acts primitively (respectively quasiprimitiv…
A famous conjecture of Lovász states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.
Let $k$ be an integer. We prove a rough structure theorem for separations of order at most $k$ in finite and infinite vertex transitive graphs. Let $G = (V,E)$ be a vertex transitive graph, let $A \subseteq V$ be a finite vertex-set with $|A| \le |V|/2$ and $|\{v \in V \setminus A : {$u \sim v$ for some $u \in A$} \}|\le k$. We show that whenever the diameter of $G$ is at least $31(k+1)^2$, either $|A| \le 2k^3+k^2$, or $G$ has a ring-like structure (with bounded parameters), and $A$ is efficiently contained in an interval. This theorem may be viewed as a rough characterization, generalizing a…
We prove that, given a finite graph $Σ$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $Σ$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.
In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $Γ$, $Γ\not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\la…
In this paper we study $G$-arc-transitive graphs $Δ$ where the permutation group $G_x^{Δ(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $Δ(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $Δ$, if $(x,y)$ is an arc of $Δ$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $Δ(x)$ and $Δ(y)$ is a $p$-group for some prime $p$. Next we prove that every $G$-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-l…
A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the tetravalent vertex-transitive non-Cayley graphs of order $6p$ are classified for each prime $p$.
Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\bigl({diam(X_n)^δ}\bigr)$ for $δ> 1$ sufficiently sma…
Let $p$ and $q$ be distinct odd primes. Let $Γ=(V(Γ), E(Γ))$ be a non-Cayley vertex-transitive graph of order $pq.$ Let $G\leq \Aut(Γ)$ acts primitively on the vertex set $V(Γ)$. In this paper, we show that $G$ is uniprimitive which is primitive but not 2-transitive and we obtain some information about $p, q$ and the minimality of the Socle $T=\soc(G).$
We prove a quantitative, finitary version of Trofimov's result that a connected, locally finite vertex-transitive graph G of polynomial growth admits a quotient with finite fibres on which the action of Aut(G) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph G of large diameter admits a quotient with fibres of small diameter on which the action of Aut(G) is virtually abelian with vertex stabilisers of bounded size. We also show that G has moderate growth in the sense of Diaconis and Saloff-Coste, …
In this article, we study eigenvalue functions of varying transition probability matrices on finite, vertex transitive graphs. We prove that the eigenvalue function of an eigenvalue of fixed higher multiplicity has a critical point if and only if the corresponding spectral representation is equilateral. We also show how the geometric realisation of a finite Coxeter group as a reflection group can be used to obtain an explicit orthogonal system of eigenfunctions. Combining both results, we describe the behaviour of the spectral representations of the second highest eigenvalue function under the…
A graph Γis said to be {\em symmetric} if its automorphism group \Aut(Γ) is transitive on the arc set of Γ. Let $G$ be a finite non-abelian simple group and let Γbe a connected pentavalent symmetric graph such that G\leq \Aut(Γ). In this paper, we show that if $G$ is transitive on the vertex set of Γ, then either G\unlhd \Aut(Γ) or \Aut(Γ) contains a non-abelian simple normal subgroup $T$ such that $G\leq T$ and $(G,T)$ is one of $58$ possible pairs of non-abelian simple groups. In particular, if $G$ is arc-transitive, then $(G,T)$ is one of $17$ possible pairs, and if $G$ is regular on the ve…
In this paper, we consider the Voronoi decompositions of an arbitrary infinite vertex-transitive graph G. In particular, we are interested in the following question: what is the largest number of Voronoi cells that must be infinite, given sufficiently (but finitely) many Voronoi sites which are sufficiently far from each other? We call this number the survival number s(G). The survival number of a graph has an alternative characterization in terms of covering, which we use to show that s(G) is always at least two. The survival number is not a quasi-isometry invariant, but it remains open wheth…
A non-complete graph $Γ$ is said to be $(G,2)$-distance transitive if $G$ is a subgroup of the automorphism group of $Γ$ that is transitive on the vertex set of $Γ$, and for any vertex $u$ of $Γ$, the stabilizer $G_u$ is transitive on the sets of vertices at distance 1 and 2 from $u$. This paper investigates the family of $(G,2)$-distance transitive graphs that are not $(G,2)$-arc transitive. Our main result is the classification of such graphs of valency not greater than 5.
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 large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgako…
