We give necessary and sufficient conditions for lobe-transitivity of locally finite and locally countable graphs whose connectivity equals 1. We show further that, given any biconnected graph Λ and a “code” assigned to each orbit of Aut(Λ), there exists a unique lobe-transitive graph Γ of connectivity 1 whose lobes are copies of Λ and is consistent with the given code at every vertex of Γ. These results lead to necessary and sufficient conditions for a graph of connectivity 1 to be edge-transitive and to be arc-transitive. Countable graphs of connectivity 1 the action of whose automorphism groups is, respectively, vertex-transitive, primitive, regular, Cayley, and Frobenius had been previously characterized in the literature.

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A graphical Frobenius representation (GFR) of a Frobenius group G is a graph whose automorphism group, as a group of permutations of the vertex set, is isomorphic to G. The problem of classifying which Frobenius groups admit a GFR is a natural extension of the classification of groups that have a graphical regular representation (GRR), which occupied many authors from 1958 through 1982. In this paper, we review for graph theorists some standard and deep results about finite Frobenius groups, determine classes of finite Frobenius groups and individual groups that do and do not admit GFRs, and classify those Frobenius groups of order at most 300 having a GFR. Because a Frobenius group, as opposed to a regular permutation group, has a highly restricted structure, the GFR problem emerges as algebraically more complex than the GRR problem. This paper concludes with some further questions and a strong conjecture.

A tessellation of the plane is face-homogeneous if for some integer $k\geq3$ there exists a cyclic sequence $\sigma=[p_0,p_1,\ldots,p_{k-1}]$ of integers $\geq3$ such that, for every face $f$ of the tessellation, the valences of the vertices incident with $f$ are given by the terms of $\sigma$ in either clockwise or counter-clockwise order. When a given cyclic sequence $\sigma$ is realizable in this way, it may determine a unique tessellation (up to isomorphism), in which case $\sigma$ is called monomorphic, or it may be the valence sequence of two or more non-isomorphic tessellations (polymorphic). A tessellation which whose faces are uniformly bounded in the Euclidean plane is called a Euclidean tessellation; a non-Euclidean tessellation whose faces are uniformly bounded in the hyperbolic plane is called hyperbolic. Hyperbolic tessellations are well-known to have exponential growth. We seek the face-homogeneous hyperbolic tessellation(s) of slowest growth and show that the least growth rate of monomorphic face-homogeneous tessellations is the golden mean, $\gamma=(1+\sqrt{5})/2$, attained by the sequences $[4,6,14]$ and $[3,4,7,4]$. A polymorphic sequence may yield non-isomorphic tessellations with different growth rates. However, all such tessellations found thus far grow at rates greater than $\gamma$.

A group A acting faithfully on a set X is 2-distinguishable if there is a 2-coloring of X that is not preserved by any nonidentity element of A, equivalently, if there is a proper subset of X with trivial setwise stabilizer. The motion of an element a in A is the number of points of X that are moved by a, and the motion of the group A is the minimal motion of its nonidentity elements. For finite A, the Motion Lemma says that if the motion of A is large enough (specifically at least 2 log_2 |A|), then the action is 2-distinguishable. For many situations where X has a combinatorial or algebraic structure, the Motion Lemma implies the action of Aut(X) on X is 2-distinguishable in all but finitely many instances. We prove an infinitary version of the Motion Lemma for countably infinite permutation groups, which states that infinite motion is large enough to guarantee 2-distinguishability. From this we deduce a number of results, including the fact that every locally finite, connected graph whose automorphism group is countably infinite is 2-distinguishable. One cannot extend the Motion Lemma to uncountable permutation groups, but nonetheless we prove that 2-distinguishable permutation groups with infinite motion are dense in the class of groups with infinite motion. We conjecture an extension of the Motion Lemma which we expect holds for a restricted class of uncountable permutation groups, and we conclude with a list of open questions. The consequences of our results are drawn for orbit equivalence of infinite permutation groups.

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

The {\em 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 {\em distinguishing number} of a graph is the distinguishing number of its full automorphism group acting on its vertex set. A connected graph $\Gamma$ is said to have {\em 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 {\em suborbit} of $G$. We prove that every connected primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any infinite, connected, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive 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.

Using results from group theory, we offer a concise proof of the imprimitivity of locally finite, vertex-transitive, 1-ended planar graphs, a result previously established by J.E. Graver and M. E. Watkins (2004) using graph-theoretical methods.

Topological properties of infinite graphs may be global or local. The number of ends (equivalence classes of rays that cannot be separated by a finite subgraph) and whether a given end contains an infinite set of pairwise disjoint rays describe an infinite graph globally. Automorphisms are of interest in terms of both the cardinalities of their set of orbits as well as the cardinalities of the orbits themselves. The notion of connectivity is refined to consider whether the deletion of a subgraph leaves finite or infinite components. The rate of growth, whether polynomial or exponential, tells much about the graph's global structure. Embedding of infinite graphs is of interest primarily in non-compact surfaces such as the plane, but even in the plane, issues arise concerning accumulation points. The interaction of these considerations is brought to bear on the structure of infinite planar graphs and maps.

A tessellation is understood to be a 1-ended, locally finite, 3-connected planar map. The edge-symbol $\langle p,q;k,\ell\rangle$ of an edge of a tessellation T is a 4-tuple listing the valences p and q of its two incident vertices and the covalences k and $\ell$ of its two incident faces. To say that T is edge-homogeneous means that all edges of T have the same edge-symbol. By a result of Grünbaum and Shephard, each edge-transitive tessellation may be identified with its edge-symbol. It is shown that the growth rate of T is given by a function $g(t)=\frac12(t-2+\sqrt{t^2-4t})$ of the single variable $t=(\frac{p+q}2-2)(\frac{k+\ell}2-2)$, except that the growth rate equals $g(t-1)$ when the edge-symbol of T or its planar dual has the form $\langle3,q;4,4\rangle$, where $q\geq6$. Thus, for each integer $t\geq4$, there are only finitely many edge-homogeneous tessellations whose growth rate equals $g(t)$, allowing a complete list of such tessellations to be compiled in terms of increasing growth rate. The maximum value of the quantity $\frac1p+\frac1q+\frac1k+\frac1\ell$ for tessellations with given value t is shown to decrease monotonically as t increases, while the minimum value decreases only asymptotically. Methods are demonstrated for concrete enumeration of the sets of faces and vertices at any given facial distance from a fixed face, edge, or vertex.

Given a cyclic d-tuple of integers at least 3, we consider the class of all 1-ended 3-connected d-valent planar maps such that every vertex manifests this d-tuple as the (clockwise or counterclockwise) cyclic order of covalences of its incident faces. We obtain necessary and/or sufficient conditions for the class to contain a Cayley map, a non-Cayley map whose underlying graph is a Cayley graph, a vertex-transitive graph whose subgroup of orientation-preserving automorphisms acts (or fails to act) vertex-transitively, a non-vertex-transitive map, or no planar map at all.

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

Let “ be an infinite, locally finite, connected graph withdistance function δ. Given a ray P in Γ and aconstant C ≥ 1, a vertex-sequence$\{{{x}}_{{n}}\}_{{{n}}={{0}}}^\infty\subseteq {{VP}}$ is said tobe regulated by C if, for all n εℕ,${{x}}_{{{n}}+{{1}}}$ never precedes x n on P , each vertex of P appears at most C times inthe sequence, and$\delta_{{P}}({{x}}_{{n}},{{x}}_{{{n}}+{{1}}})\leq {{C}}$. R. Halin(Math. Ann., 157, [1964], 125137) defined two rays to be end-equivalent if they are joined by infinitely manypairwise-disjoint paths; the resulting equivalence classes arecalled ends . More recently H. A. Jung (Graph StructureTheory, Contemporary Mathematics, 147, [1993], 477484) defined rays P and Q to be b-equivalent if there existsequences $\{{{x}}_{{n}}\}_{{{n}}={{0}}}^\infty\subseteq {{VP}}$and $\{{{y}}_{{n}}\}_{{{n}}={{0}}}^\infty\subseteq {{VQ}}$ VQ regulated by some constant C e 1 such that$\delta({{x}}_{{n}},{{y}}_{{n}})\leq {{C}}$ for all n εℕ; he named the resulting equivalenceclasses b-fibers . Let $F_0$ denote the set of nondecreasingfunctions from $N$ into the set of positive real numbers. Therelation ${{P}}\sim_{{f}} {{Q}}$ (called f-equivalence )generalizes Jung's condition to$\delta({{x}}_{{n}},{{y}}_{{n}})\leq {{Cf}}({{n}})$. As f runs through $\cal{F}_{{0}}$, uncountably many equivalencerelations are produced on the set of rays that are no finer than b -equivalence while, under specified conditions, are nocoarser than end-equivalence. Indeed, for every “ thereexists an "end-defining function" ${{f}}\in F_{{0}}$ that isunbounded and sublinear and such that ${{P}}\sim_{{f}} {{Q}}$implies that P and Q are end-equivalent. Say${{P}}\approx {{Q}}$ if there exists a sublinear function ${{f}}\inF_{{0}}$ such that ${{P}}\sim_{{f}} {{Q}}$. The equivalence classeswith respect to $\approx$ are called bundles . We pursue thenotion of "initially metric" rays in relation to bundles, and showthat in any bundle either all or none of its rays are initiallymetric. Furthermore, initially metric rays in the same bundle areend-equivalent. In the case that Γ contains translatable rayswe give some sufficient conditions for every f -equivalenceclass to contain uncountably many g -equivalence classes(where ${lim}_{{{n}}\to\infty}{{g}}({{n}})/{{f}}({{n}})={{0}}$). Weconclude with a variety of applications to infinite planar graphs.Among these, it is shown that two rays whose union is the boundaryof an infinite face of an almost-transitive planar map are neverbundle- equivalent. © 2006 Wiley Periodicals, Inc. J GraphTheory 54: 125153, 2007

We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the Institut Henri Poincare in December 2004.

A Bilinski diagram (respectively, B∗-diagram) is a labeling of a planar map with respect to the regional distance of its vertices and faces from a central vertex (respectively, face). Such diagrams are concentric if for each k ≥ 1, the set of vertices at regional distance k from the central vertex or face induces a circuit. The class Ga,b consists of all 1-ended, 3-connected planar maps with the property that every valence is finite and at least a and every covalence is finite and at least b. A map in the subclass Ga,b+ of Ga,b contains no adjacent b-covalent faces, and dually a map in Ga+,b contains no adjacent a-valent vertices. It is shown that all Bilinski diagrams and all B∗-diagrams of all maps in G6,3, G4,4, G3,6, G5,3+ and G3+,5 are concentric.

What do you do to start reading locally finite planar edge transitive graphs? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this locally finite planar edge transitive graphs.

These lectures introduce the finite graph theorist to a medley of topics and theorems in infinite graphs theory. Section 1: three graph theoretical notions required for a study of infinite graphs, namely end-equivalence (as developed by R. Halin), a refinement of the notion of connectivity, and growth. Section 2: an extension to infinite graphs (by C. Thomassen and the author) of W.T. Tutte's thoerem on arc-transitivity. Section 3: two-ended graphs, especially various characterizations of strips. Section 4: rays, double rays, quasi-axes, and the automorphism group action upon them. Section 5: joint work by P. Niemeyer and the author on fiber-equivalence, which is a refinement of end-equivalence. Section 6: the classification of locally finite, edge-transitive planar graphs by J.E. Graver and the author in terms of the number of ends, their Petrie walks, and the local behavior of their automorphism groups.

Let τ be an infinite graph, let π be a double ray in τ, and letd anddπ denote the distance functions in τ and in π, respectively. One calls π anaxis ifd(x,y)=d π (x,y) and aquasi-axis if lim infd(x,y)/d π (x,y)>0 asx, y range over the vertex set of π andd π (x,y)→∞. The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if α is a translation whose directionD(α) is a thin end, then there exists an axis inD(α) andD(α−1) invariant under α r for somer not exceeding the maximum number of disjoint rays inD(α).The thinness ofD(α) is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.