Vertex-symmetric Graphs Research Articles

Hamiltonian paths in vertex-symmetric graphs of order 5p

It is shown that every connected vertex-symmetric graph of order 5p (p a prime) has a Hamiltonian path.

On vertex symmetric digraphs

It is proved that if p is a prime, k and m⩽ p are positive integers, and I is a vertex symmetric diagraph of order p k or mp, then Γ has an automorphism all of whose orbits have cardinality p. Vertex symmetric graphs of order 2 p such that 2 p−1 is not the square of a composite integer and vertex symmetric digraphs of order p k are characterised.

On vertex symmetric digraphs

It is proved that if p is a prime, k and m ⩽ P are positive integers, and r is a vertex symmetric digraph of order pk or mp, then r has an automorphism all of whose orbits have cardinality p. Vertex symmetric graphs of order 2p such that 2p 1 is not the square of a composite integer and vertex symmetric digraphs of order pk are characterised.

General topological results on the construction of a minimum essential set of a directed graph

In this paper we present general topological results on the construction of a minimum essential set or minimum feedback vertex set of a directed graph. The results include all the existing topological rules that identify subsets of a minimum essential set as special cases. Moreover, we show how a class of topological results can be systematically generated by using the theory of strongly adjacent polygons. We use the topological results to provide an algorithm which constructs a minimal essential set of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -vertex symmetric directed graph, a maximal stable set of an n-vertex undirected graph and an essential set of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -vertex arbitrary directed graph in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta (n^2)</tex> computation steps and such that these solutions are within a known tolerance of the optimal value.

FREE PRODUCTS OF NETWORKS AND FREE SYMMETRIZERS OF GRAPHS

Connected locally finite vertex-symmetric graphs, also called networks, are the subject of the article. The operation of a free product, preserving the vertex symmetry of graphs, is introduced. Properties of free products are studied, and a connection with free products of groups is established. A class of networks, called free symmetrizers of graphs, in some respects analogues of free products, is introduced. Their properties and a relation to free products and graphs of groups are studied.Figures: 3. Bibliography: 12 titles.