Almost simple groups of Suzuki type acting on polytopes

Let , with an odd power of two. For each almost simple group such that , we prove that is not a C-group and therefore is not the automorphism group of an abstract regular polytope. For , we show that there is always at least one abstract regular polytope such that . Moreover, if is an abstract regular polytope such that , then is a polyhedron.

     

Affine distance-transitive graphs and classical groups

This paper finishes the classification of the finite primitive affine distance-transitive graphs by dealing with the only case left open, namely where the generalized Fitting subgroup of the stabilizer of a vertex is modulo scalars a simple group of classical Lie type defined over the characteristic dividing the number of vertices of the graph. All graphs that are found to occur are known.     

The locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

In this paper, seven families of vertex-intransitive locally (G,2)-arc transitive graphs are constructed, where Sz(q)?G?Aut(Sz(q)), q=²2k+1 for some k∈N. It is then shown that for any graph Γ in one of these families, Sz(q)?Aut(Γ)?Aut(Sz(q)) and that the only locally 2-arc transitive graphs admitting an almost simple group of Suzuki type whose vertices all have valency at least three are (i) graphs in these seven families, (ii) (vertex transitive) 2-arc transitive graphs admitting an almost simple group of Suzuki type, or (iii) double covers of the graphs in (ii). Since the graphs in (ii) have been classified by Fang and Praeger (1999) [6], this completes the classification of locally 2-arc transitive graphs admitting a Suzuki simple group     

Free groups and ends of graphs

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms are discussed.The set of ends which are fixed by some hyperbolic element of a group G acting on X is denoted by ?(G). If G contains a hyperbolic automorphism of X and G fixes no end of X, then G contains a free subgroup F such that ?(F) is dense in ?(G) with respect to the natural topology on the ends of X.As an application we obtain the following: A group which acts transitively on a connected graph and fixes no end has a free subgroup whose directions are dense in the end boundary.     

Partition graphs for finite symmetric groups

This paper outlines an investigation of a class of arc-transitive graphs admitting a finite symmetric group S_n acting primitively on vertices, with vertex-stabilizer isomorphic to the wreath product S_m wr S_r (preserving a partition of {1,2,…n} into r parts of equal size m). Several properties of these graphs are considered, including their correspondence with r × r matrices with constant row- and column-sums equal to m, their girth, and the local action of the vertex-stabilizer. Also, it is shown that the only instance where S_n acts transitively on 2-arcs occurs in the case m = r = 2. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 107–117, 1997     

Isomorphic Cayley graphs on nonisomorphic groups

The issue of when two Cayley digraphs on different abelian groups of prime power order can be isomorphic is examined. This had previously been determined by Anne Joseph for squares of primes; her results are extended. © 1999 John Wiley & Sons, Inc. J Graph Theory 3: 345–362, 1999     

Normal and non-normal Cayley graphs for symmetric groups

A Cayley graph is said to be an NNN-graph if its automorphism group contains two isomorphic regular subgroups where one is normal and the other is non-normal. In this paper, we show that there exist NNN-graphs among the Cayley graphs for symmetric groups $S_n$ if and only if $n?5$.     

A large family of cospectral Cayley graphs over dicyclic groups

For a finite group G and an inverse closed subset $S\subseteq G\setminus \{e\}$, the Cayley graph $X(G,S)$ has vertex set G and two vertices $x,y\in G$ are adjacent if and only if $x{y}^{-1}\in S$. Two graphs are called cospectral if their adjacency matrices have the same spectrum. Let $p\ge 3$ be a prime number and $T_{4p}$ be the dicyclic group of order 4p. In this paper, with the help of the characters from representation theory, we construct a large family of pairwise non-isomorphic and cospectral Cayley graphs over the dicyclic group $T_{4p}$ with $p\ge 23$, and find several pairs of non-isomorphic and cospectral Cayley graphs for $5\le p\le 19$.     

Planarity of Cayley graphs of graph products of groups

We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.     

On extremal k-outconnected graphs

Let G be a minimally k-connected graph with n nodes and m edges. Mader proved that if n?3k-2 then m?k(n-k), and for n?3k-1 an equality is possible if, and only if, G is the complete bipartite graph K_k,n-k. Cai proved that if n?3k-2 then m?⌊(n+k)²/8⌋, and listed the cases when this bound is tight.In this paper we prove a more general theorem, which implies similar results for minimally k-outconnected graphs; a graph is called k-outconnected from r if it contains k internally disjoint paths from r to every other node.     

The maximum number of maximal independent sets in unicyclic connected graphs

In this paper, we determine the maximum number of maximal independent sets in a unicyclic connected graph. We also find a class of graphs achieving this maximum value.     

Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees

The first Zagreb index M₁(G) and the second Zagreb index M₂(G) of a (molecular) graph G are defined as M₁(G)=∑_u∈V(G)(d(u))² and M₂(G)=∑_uv∈E(G)d(u)d(v), where d(u) denotes the degree of a vertex u in G. The AutoGraphiX system [M. Aouchiche, J.M. Bonnefoy, A. Fidahoussen, G. Caporossi, P. Hansen, L. Hiesse, J. Lacheré, A. Monhait, Variable neighborhood search for extremal graphs. 14. The AutoGraphiX 2 system, in: L. Liberti, N. Maculan (Eds.), Global Optimization: From Theory to Implementation, Springer, 2005; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system, Discrete Math. 212 (2000) 29-44; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures, Discrete Math. 276 (2004) 81-94] conjectured that M₁/n≤M₂/m (where n=|V(G)| and m=|E(G)|) for simple connected graphs. Hansen and Vuki?evi? [P. Hansen, D. Vuki?evi?, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168] proved that it is true for chemical graphs and it does not hold for all graphs. Vuki?evi? and Graovac [D. Vuki?evi?, A. Graovac, Comparing Zagreb M₁ and M₂ indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57 (2007) 587-590] proved that it is also true for trees. In this paper, we show that M₁/n≤M₂/m holds for graphs with Δ(G)−δ(G)≤2 and characterize the extremal graphs, the proof of which implies the result in [P. Hansen, D. Vuki?evi?, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168]. We also obtain the result that M₁/n≤M₂/m holds for graphs with Δ(G)−δ(G)≤3 and δ(G)≠2.     

The fine structure of octahedron-free graphs

This paper is one of a series of papers in which, for a family L of graphs, we describe the typical structure of graphs not containing any L∈L. In this paper, we prove sharp results about the case L={O₆}, where O₆ is the graph with 6 vertices and 12 edges, given by the edges of an octahedron. Among others, we prove the following results.(a) The vertex set of almost every O₆-free graph can be partitioned into two classes of almost equal sizes, U₁ and U₂, where the graph spanned by U₁ is a C₄-free and that by U₂ is P₃-free.(b) Similar assertions hold when L is the family of all graphs with 6 vertices and 12 edges.(c) If H is a graph with a color-critical edge and χ(H)=p+1, then almost every sH-free graph becomes p-chromatic after the deletion of some s−1 vertices, where sH is the graph formed by s vertex disjoint copies of H.These results are natural extensions of theorems of classical extremal graph theory. To show that results like those above do not hold in great generality, we provide examples for which the analogs of our results do not hold.     

Dynamic list coloring of bipartite graphs

A dynamic coloring of a graph is a proper coloring of its vertices such that every vertex of degree more than one has at least two neighbors with distinct colors. The least number of colors in a dynamic coloring of G, denoted by χ₂(G), is called the dynamic chromatic number of G. The least integer k, such that if every vertex of G is assigned a list of k colors, then G has a proper (resp. dynamic) coloring in which every vertex receives a color from its own list, is called the choice number of G, denoted by ch(G) (resp. the dynamic choice number, denoted by ch₂(G)). It was recently conjectured (Akbari et al. (2009) [1]) that for any graph G, ch₂(G)=max(ch(G),χ₂(G)). In this short note we disprove this conjecture. We first give an example of a small planar bipartite graph G with ch(G)=χ₂(G)=3 and ch₂(G)=4. Then, for any integer k≥5, we construct a bipartite graph G_k such that ch(G_k)=χ₂(G_k)=3 and ch₂(G)≥k.