Symmetry of iteration graphs

Generalized symmetry of graphs

Symmetry of graphs has been extensively studied over the past 50 years by using automorphisms of graphs and group theory which have played an important role in graph, and promising and interesting results have been obtained[2,20,21]. We introduced generalized symmetry of graphs and investigated it by using endomorphisms of graphs and semigroup theory. In this paper, I will survey some results w...

Symmetry Breaking in Graphs

A labeling of the vertices of a graph G, φ : V (G) → {1, . . . , r}, is said to be r-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by D(G), is the minimum r such that G has an r-distinguishing labeling. The distinguishing number of the complete graph on t vertices is t. In contrast, we prove (i) given any...

Medial Symmetry Type Graphs

A k-orbit map is a map with its automorphism group partitioning the set of flags into k orbits. Recently k-orbit maps were studied by Orbanić, Pellicer and Weiss, for k 6 4. In this paper we use symmetry type graphs to extend such study and classify all the types of 5-orbit maps, as well as all self-dual, properly and improperly, symmetry type of k-orbit maps with k 6 7. Moreover, we determine,...

Visualisation of Symmetry in Graphs

Higher Order Symmetry of Graphs

This lecture gives background to and results of work of my student John Shrimpton [19, 20, 21]. It advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. Groups are expected to be associated with symmetry. Klein’s famous Erlanger Programme asserted that the study of a geometry was the study of the group of automorphisms of that geometry. ...