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Destroying symmetry by orienting edges: complete graphs and complete bigraphs

100%
Harary F., Jacobson M.
Discussiones Mathematicae Graph Theory
|
2001
|
tom 21
|
nr 2
149-158
EN
Our purpose is to introduce the concept of determining the smallest number of edges of a graph which can be oriented so that the resulting mixed graph has the trivial automorphism group. We find that this number for complete graphs is related to the number of identity oriented trees. For complete bipartite graphs $K_{s,t}$, s ≤ t, this number does not always exist. We determine for s ≤ 4 the values of t for which this number does exist.
2
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On roots of the automorphism group of a circular domain in $ℂ^n$

61%
Myszewski J.
Annales Polonici Mathematici
|
1991
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tom 55
|
nr 1
269-276
EN
We study the properties of the group Aut(D) of all biholomorphic transformations of a bounded circular domain D in $ℂ^n$ containing the origin. We characterize the set of all possible roots for the Lie algebra of Aut(D). There exists an n-element set P such that any root is of the form α or -α or α-β for suitable α,β ∈ P.
3
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Ultra regular covering space and its automorphism group

51%
Han S.-E.
International Journal of Applied Mathematics and Computer Science
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2010
|
tom 20
|
nr 4
699-710
EN
In order to classify digital spaces in terms of digital-homotopic theoretical tools, a recent paper by Han (2006b) (see also the works of Boxer and Karaca (2008) as well as Han (2007b)) established the notion of regular covering space from the viewpoint of digital covering theory and studied an automorphism group (or Deck's discrete transformation group) of a digital covering. By using these tools, we can calculate digital fundamental groups of some digital spaces and classify digital covering spaces satisfying a radius 2 local isomorphism (Boxer and Karaca, 2008; Han, 2006b; 2008b; 2008d; 2009b). However, for a digital covering which does not satisfy a radius 2 local isomorphism, the study of a digital fundamental group of a digital space and its automorphism group remains open. In order to examine this problem, the present paper establishes the notion of an ultra regular covering space, studies its various properties and calculates an automorphism group of the ultra regular covering space. In particular, the paper develops the notion of compatible adjacency of a digital wedge. By comparing an ultra regular covering space with a regular covering space, we can propose strong merits of the former.
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