scholarly journals Infinite Graphical Frobenius Representations

10.37236/7294 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Mark E. Watkins
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Polynomial Growth ◽  
Free Products ◽  
Locally Finite ◽  
Finitely Generated Groups ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Planar Maps ◽  
Transitive Graphs

A graphical Frobenius representation (GFR) of a Frobenius (permutation) group $G$ is a graph $\Gamma$ whose automorphism group Aut$(\Gamma)$ acts as a Frobenius permutation group on the vertex set of $\Gamma$, that is, Aut$(\Gamma)$ acts vertex-transitively with the property that all nonidentity automorphisms fix either exactly one or zero vertices and there are some of each kind. The set $K$ of all fixed-point-free automorphisms together with the identity is called the kernel of $G$. Whenever $G$ is finite, $K$ is a regular normal subgroup of $G$ (F. G. Frobenius, 1901), in which case $\Gamma$ is a Cayley graph of $K$. The same holds true for all the infinite instances presented here.Infinite, locally finite, vertex-transitive graphs can be classified with respect to (i) the cardinality of their set of ends and (ii) their growth rate. We construct families of infinite GFRs for all possible combinations of these two properties. There exist infinitely many GFRs with polynomial growth of degree $d$ for every positive integer $d$, and there exist infinite families of GFRs of exponential growth, both $1$-ended and infinitely-ended, that underlie infinite chiral planar maps. There also exist GFRs of free products of finitely many finitely generated groups. Graphs of connectivity 1 having a Frobenius automorphism group are characterized.

Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

2008 ◽  
Vol 15 (03) ◽  
pp. 379-390 ◽  
Author(s):  
Xuesong Ma ◽  
Ruji Wang
Keyword(s):  
Automorphism Group ◽  
Bipartite Graphs ◽  
Type Ii ◽  
Symmetric Graph ◽  
Trivalent Graph ◽  
Block Graphs ◽  
Group A ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

scholarly journals Counting subgraphs in fftp graphs with symmetry

2019 ◽  
pp. 1-27
Author(s):  
YAGO ANTOLÍN
Keyword(s):  
Generating Functions ◽  
Hyperbolic Groups ◽  
Infinite Index ◽  
Convex Subgroup ◽  
Locally Finite ◽  
Fellow Traveler ◽  
Generating Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Abstract Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano–Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications concerns Schreier coset graphs of hyperbolic groups relative to quasi-convex subgroups, we show that these graphs have rational growth, the falsification by fellow traveler property, and the existence of a lower bound for the growth rate independent of the finite generating set and the infinite index quasi-convex subgroup.

scholarly journals Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

2014 ◽  
Vol 157 (1) ◽  
pp. 45-61
Author(s):  
PABLO SPIGA
Keyword(s):  
Automorphism Group ◽  
Positive Solution ◽  
Cayley Graph ◽  
Burnside Problem ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Restricted Burnside Problem ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractIn this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

AUTOMORPHISM GROUPS OF SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS

2015 ◽  
Vol 93 (2) ◽  
pp. 238-247
Author(s):  
ZHAOHONG HUANG ◽  
JIANGMIN PAN ◽  
SUYUN DING ◽  
ZHE LIU
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Automorphism Groups ◽  
Composition Factor ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs ◽  
Free Order

Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has $\text{A}_{5}$ as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of $4$-power-free order.

scholarly journals Groups with a (B, N)–pair and locally transitive graphs

1979 ◽  
Vol 74 ◽  
pp. 1-21 ◽  
Author(s):  
Richard Weiss
Keyword(s):  
Permutation Group ◽  
Undirected Graph ◽  
Vertex Set ◽  
Transitive Graphs

Let Γ be an undirected graph and G a subgroup of aut (Γ). We denote by ∂(x, y) the distance between two vertices x and y, by E(Γ) the edge set of Γ, by V(Γ) the vertex set of Γ, by Γ(x) the set of neighbors of the vertex x and by G(x)Γ(x) the permutation group induced by the stabilizer G(x) on Γ(x).

scholarly journals FINITE VORONOI DECOMPOSITIONS OF INFINITE VERTEX TRANSITIVE GRAPHS

2013 ◽  
Vol 05 (02) ◽  
pp. 239-250
Author(s):  
HILARY FINUCANE
Keyword(s):  
Exponential Growth ◽  
Polynomial Growth ◽  
Transitive Graph ◽  
Voronoi Cells ◽  
Alternative Characterization ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

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 the number of balls of radius r-1 required to cover a sphere of radius r. The survival number is not a quasi-isometry invariant, but it remains open whether finiteness of s(G) is. We show that all vertex-transitive graphs with polynomial growth have finite s(G); vertex-transitive graphs with infinitely many ends have infinite s(G); the lamplighter graph LL(Z), which has exponential growth, has finite s(G); and the lamplighter graph LL(Z2), which is Liouville, has infinite s(G).

Local Expansion of Symmetrical Graphs

1992 ◽  
Vol 1 (1) ◽  
pp. 1-11 ◽  
Author(s):  
László Babai ◽  
Mario Szegedy
Keyword(s):  
Automorphism Group ◽  
Finite Subset ◽  
Harmonic Mean ◽  
Expansion Rate ◽  
Transitive Graph ◽  
Local Expansion ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Vertex Transitive Graph

A graph is vertex-transitive (edge-transitive) if its automorphism group acts transitively on the vertices (edges, resp.). The expansion rate of a subset S of the vertex set is the quotient e(S):= |∂(S)|/|S|, where ∂(S) denotes the set of vertices not in S but adjacent to some vertex in S. Improving and extending previous results of Aldous and Babai, we give very simple proofs of the following results. Let X be a (finite or infinite) vertex-transitive graph and let S be a finite subset of the vertices. If X is finite, we also assume |S| ≤|V(X)/2. Let d be the diameter of S in the metric induced by X. Then e(S) ≥1/(d + 1); and e(S) ≥ 2/(d +2) if X is finite and d is less than the diameter of X. If X is edge-transitive then |δ(S)|/|S| ≥ r/(2d), where ∂(S) denotes the set of edges joining S to its complement and r is the harmonic mean of the minimum and maximum degrees of X. – Diverse applications of the results are mentioned.

On quasiprimitive edge-transitive graphs of odd order and twice prime valency

2020 ◽  
Vol 23 (6) ◽  
pp. 1017-1037
Author(s):  
Hong Ci Liao ◽  
Jing Jian Li ◽  
Zai Ping Lu
Keyword(s):  
Automorphism Group ◽  
Connected Graph ◽  
Automorphism Groups ◽  
Primitive Group ◽  
Odd Order ◽  
Vertex Set ◽  
Edge Transitive ◽  
Affine Type ◽  
Transitive Graphs

AbstractA 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 {\varGamma} be a connected graph of odd order and twice prime valency, and let G be a subgroup of the automorphism group of {\varGamma}. In the case where G acts transitively on the edge set and quasiprimitively on the vertex set of {\varGamma}, we prove that either G is almost simple, or G is a primitive group of affine type. If further G is an almost simple primitive group, then, with two exceptions, the socle of G acts transitively on the edge set of {\varGamma}.

scholarly journals Distinguishability of Infinite Groups and Graphs

10.37236/2283 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Simon M Smith ◽  
Thomas W Tucker ◽  
Mark E Watkins
Keyword(s):  
Finite Graph ◽  
Primitive Permutation Group ◽  
Infinite Groups ◽  
Cartesian Products ◽  
Distinguishing Number ◽  
Locally Finite Graph ◽  
Point Stabilizer ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Transitive Graphs

The distinguishing number of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph $\Gamma$ is said to have connectivity 1 if there exists a vertex $\alpha \in V\Gamma$ such that $\Gamma \setminus \{\alpha\}$ is not connected. For $\alpha \in V$, an orbit of the point stabilizer $G_\alpha$ is called a suborbit of $G$.We prove that every nonnull, primitive graph with infinite diameter and countably many vertices has distinguishing number $2$. Consequently, any nonnull, infinite, primitive, locally finite graph is $2$-distinguishable; so, too, is any infinite primitive permutation group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number $2$. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma.

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