Isomorphisms and automorphisms of graph coverings

Hofmeister, M.

doi:10.1016/0012-365x(91)90374-b

Cited by 27 publications

(10 citation statements)

References 9 publications

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“…5 . N ON -ISOMORPHIC S WITCHING E QUIVALENCE C LASSES In this section we will develop a counting formula for the number of non-isomorphic switching equivalence classes of the graph G .…”

Section: T Heorem 2 the Number Of Non -Isomorphic Ertex Colorings Omentioning

confidence: 95%

“…For more details , see , for example [5] . An interesting application of counting graph coverings can be found in [9] .…”

Section: A T Opological a Pplicationmentioning

confidence: 99%

“…If χ ( ‫ކ‬ q ) ϶ 2 , then the value of F on diagonal arcs is given by equation (6) . The other arc values of F are then determined recursively by the fact that F has to be alternating and by equation (5) . Equation (4) guarantees the consistency of the procedure .…”

Section: T Heorem 2 the Number Of Non -Isomorphic Ertex Colorings Omentioning

confidence: 99%

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On an Exact Sequence Related to a Graph

Hofmeister

1996

European Journal of Combinatorics

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The five problems of counting component colorings , vertex colorings , arc colorings , cocycles and switching equivalence classes of a graph with respect to a finite field up to isomorphism are related by an exact sequence that stems from a coboundary operator . This cohomology is presented , and counting formulas are given for each of the five problems . Finally , a topological application is given .÷ 1996 Academic Press Limitedand automorphism group ⌫ . Two objects related to G (e . g . vertices , edges , components , vertex colorings , etc . ) are called isomorphic , if there is an automorphism of G mapping the one onto the other .For some prime power q , let ‫ކ‬ q be the finite field of this order . Consider the following problems , which are related by the common use of the field ‫ކ‬ q . There is a nice cohomological approach relating these five problems . Our purpose is to present this cohomology and solutions of the problems . All the solutions are based on Burnside's lemma or its famous implication , Po ´ lya's theorem . Because of their importance , we will present these classical results in the next section .As it will be seen , Problems 1 and 2 can be easily solved by Po ´ lya's theorem .Problem 3 reduces to the problem of coloring the edges of G with q colors up to isomorphism if the field characteristic of ‫ކ‬ q (denoted by χ ( ‫ކ‬ q )) equals two ; in this case , the enumeration can be done by Po ´ lya's theorem again . Problem 4 is solved in [6] in full generality ; therefore we will present only the result , but not the proof .

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“…5 . N ON -ISOMORPHIC S WITCHING E QUIVALENCE C LASSES In this section we will develop a counting formula for the number of non-isomorphic switching equivalence classes of the graph G .…”

Section: T Heorem 2 the Number Of Non -Isomorphic Ertex Colorings Omentioning

confidence: 95%

“…For more details , see , for example [5] . An interesting application of counting graph coverings can be found in [9] .…”

Section: A T Opological a Pplicationmentioning

confidence: 99%

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On an Exact Sequence Related to a Graph

Hofmeister

1996

European Journal of Combinatorics

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“…These include: counting isomorphism classes of coverings and, more generally, graph bundles, as considered by Hofmeister [15] and Kwak and Lee [17,18]; constructions of regular maps on surfaces based on covering space techniques due to Archdeacon, Gvozdjak, Nedela, Richter,Širáň,Škoviera and Surowski [1,2,14,28,29,37]; and construction of transitive graphs with a prescribed degree of symmetry, for instance by Du, Malnič, Nedela, Marušič, Scapellato, Seifter, Trofimov and Waller [8,22,23,25,26,34]. Lifting and/or projecting techniques play a prominent role also in the study of imprimitive graphs, cf.…”

Section: Historymentioning

confidence: 99%

Lifting Graph Automorphisms by Voltage Assignments

Malnič

Nedela

Škoviera

2000

European Journal of Combinatorics

152

145

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The problem of lifting graph automorphisms along covering projections and the analysis of lifted groups is considered in a purely combinatorial setting. The main tools employed are: (1) a systematic use of the fundamental groupoid; (2) unification of ordinary, relative and permutation voltage constructions into the concept of a voltage space; (3) various kinds of invariance of voltage spaces relative to automorphism groups; and (4) investigation of geometry of the lifted actions by means of transversals over a localization set. Some applications of these results to regular maps on surfaces are given. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful re-examination of the whole subject and at the same time leads to simplification and generalization of several known results.

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“…Another, but quite similar, approach for constructing covering maps uses so-called "voltage graph" [7][8][9]. Voltage graphs and G-chains are strongly related.…”

Section: Constructing Covering Graphs From Chain Mapsmentioning

confidence: 99%

Towards a Calculus of Biological Networks

Reidys¹,

Mortveit²

2002

Zeitschrift Für Physikalische Chemie

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In this paper we present a new framework for studying the dynamics of biological networks. A specific class of dynamical systems, Sequential Dynamical Systems (SDS), is introduced. These systems allow one to investigate the interplay between structural properties of the network and its phase space. We will show in detail how to find a reduced system that captures key features of a given system. This reduction is based on a special graph-theoretic relation between the two networks. We will study the reduction of SDS over n-cubes in detail and we will present several examples.

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Isomorphisms and automorphisms of graph coverings

Cited by 27 publications

References 9 publications

On an Exact Sequence Related to a Graph

On an Exact Sequence Related to a Graph

Lifting Graph Automorphisms by Voltage Assignments

Towards a Calculus of Biological Networks

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