Vertex-primitive groups and graphs of order twice the product of two distinct odd primes

A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertices which is not a Cayley graph. In this paper, we complete the determination of the non-Cayley numbers of the form 2pq, where p, q are distinct odd primes. Earlier work of Miller and the second author had dealt with all such numbers corresponding to vertex-transitive graphs admitting an imprimitive subgroup of automorphisms. This paper deals with the primitive case. First the primitive permutation groups of degree 2pq are classified. This depends on the finite simple group classification. Then each of these groups G is examined to determine whether there are any non-Cayley graphs which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups. The outcome is that 2pq is a non-Cayley number, where

Some Problems in Algebraic and Extremal Graph Theory.

In this dissertation, we consider a wide range of problems in algebraic and extremal graph theory. In extremal graph theory, we will prove that the Tree Packing Conjecture is true for all sequences of trees that are \u27almost stars\u27; and we prove that the Erdos-Sos conjecture is true for all graphs G with girth at least 5. We also conjecture that every graph G with minimal degree k and girth at least $2t+1$ contains every tree T of order $kt+1$ such that $\Delta(T)\leq k.$ This conjecture is trivially true for t = 1. We Prove the conjecture is true for t = 2 and that, for this value of t, the conjecture is best possible. We also provide supporting evidence for the conjecture for all other values of t. In algebraic graph theory, we are primarily concerned with isomorphism problems for vertex-transitive graphs, and with calculating automorphism groups of vertex-transitive graphs. We extend Babai\u27s characterization of the Cayley Isomorphism property for Cayley hypergraphs to non-Cayley hypergraphs, and then use this characterization to solve the isomorphism problem for every vertex-transitive graph of order pq, where p and q distinct primes. We also determine the automorphism groups of metacirculant graphs of order pq that are not circulant, allowing us to determine the nonabelian groups of order pq that are Burnside groups. Additionally, we generalize a classical result of Burnside stating that every transitive group G of prime degree p, is doubly transitive or contains a normal Sylow p-subgroup to all p\sp k, provided that the Sylow p-subgroup of G is one of a specified family. We believe that this result is the most significant contained in this dissertation. As a corollary of this result, one easily gives a new proof of Klin and Poschel\u27s result characterizing the automorphism groups of circulant graphs of order p\sp k, where p is an odd prime

Vertex-transitive digraphs with extra automorphisms that preserve the natural arc-colouring

Diamond open accessIn a Cayley digraph on a group G, if a distinct colour is assigned to each arc-orbit under the left-regular action of G, it is not hard to show that the elements of the left-regular action of G are the only digraph automorphisms that preserve this colouring. In this paper, we show that the equivalent statement is not true in the most straightforward generalisation to G-vertex-transitive digraphs, even if we restrict the situation to avoid some obvious potential problems. Specifically, we display an infinite family of 2-closed groups G, and a G-arc-transitive digraph on each (without any digons) for which there exists an automorphism of the digraph that is not an element of G (it is an automorphism of G). Since the digraph is G-arc-transitive, the arcs would all be assigned the same colour under the colouring by arc-orbits, so this digraph automorphism is colour-preserving.Ye

AUTOMORPHISM GROUPS OF MAPS, SURFACES AND SMARANDACHE GEOMETRIES

Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automorphisms simply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page