scholarly journals Enumeration of Unlabeled Directed Hypergraphs

10.37236/2766 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Jianguo Qian
Keyword(s):  
Permutation Group ◽  
Equivalence Class ◽  
Directed Graphs ◽  
Disjoint Cycle ◽  
Cycle Decomposition ◽  
Cycle Index ◽  
Index Method ◽  
Directed Hypergraphs ◽  
Counting Lemma

We consider the enumeration of unlabeled directed hypergraphs by using Pólya's counting theory and Burnside's counting lemma. Instead of characterizing the cycle index of the permutation group acting on the hyperarc set $A$, we treat each cycle in the disjoint cycle decomposition of a permutation $\rho$ acting on $A$ as an equivalence class (or orbit) of $A$ under the operation of the group generated by $\rho$. Compared to the cycle index method, our approach is  more effective in  dealing with the enumeration of directed hypergraphs. We deduce the explicit counting formulae for the unlabeled $q$-uniform and unlabeled general directed hypergraphs. The former generalizes the well known result for 2-uniform directed hypergraphs, i.e., for the ordinary directed graphs introduced by Harary and Palmer.

scholarly journals Application of GAP (groups, algorithms and programming) system to stereoisograms for characterizing RS-stereoisomers of cubane derivatives

2021 ◽  
Vol 87 (2) ◽  
pp. 207-270
Author(s):  
Shinsaku Fujita ◽  
Keyword(s):  
Permutation Group ◽  
Point Group ◽  
Gap Functions ◽  
Programming System ◽  
Cycle Index ◽  
Index Method ◽  
Type V ◽  
Partial Cycle

The PCI (Partial-Cycle-Index) method of Fujita’s USCI (Unit-Subduced-CycleIndex) approach has been applied to symmetry-itemized enumerations of cubane derivatives, where groups for specifying three-aspects of symmetry, i.e., the point group for chirality/achirality, the RS-stereogenic group for RS-stereogenicity/RS-astereogenicity, and the LR-permutation group for sclerality/ascrelarity are considered as the subgroups of the RS-stereoisomeric group . Five types of stereoisograms are adopted as diagrammatical expressions of , after combined-permutation representations (CPR) are created as new tools for treating various groups according to Fujita’s stereoisogram approach. The use of CPRs under the GAP (Groups, Algorithms and Programming) system has provided new GAP functions for promoting symmetry-itemized enumerations. The type indices for characterizing stereoisograms (e.g., for a type-V stereoisogram) have been sophisticated into RS-stereoisomeric indices (e.g., for a cubane derivative with the composition ). The type-V stereoisograms for cubane derivatives with the composition are discussed under extended pseudoasymmetry as a new concept.

Self-Complementary Generalized Orbits of a Permutation Group

1974 ◽  
Vol 17 (2) ◽  
pp. 203-208 ◽  
Author(s):  
Roberto Frucht ◽  
Frank Harary
Keyword(s):  
Permutation Group ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Cycle Index ◽  
Arbitrary Permutation ◽  
Group A ◽  
Special Classes

AbstractA permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k) the set of all k subsets of X, and the resulting equivalence classes are called k orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X—S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits.

scholarly journals Permutation group algorithms based on directed graphs

2021 ◽  
Author(s):  
Christopher Jefferson ◽  
Markus Pfeiffer ◽  
Wilf A. Wilson ◽  
Rebecca Waldecker
Keyword(s):  
Permutation Group ◽  
Directed Graphs

scholarly journals Cycle Index, Weight Enumerator, and Tutte Polynomial

10.37236/1663 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Peter J. Cameron
Keyword(s):  
Permutation Group ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Weight Enumerator ◽  
Permutation Groups ◽  
Cycle Index ◽  
Transitive Groups ◽  
Index Weight

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a group from a code, the matroid differs only trivially from that which arises directly from the code. In this case, the Tutte polynomial of the code specialises to the weight enumerator (by Greene's Theorem), and hence also to the cycle index. However, in another subclass of IBIS groups, the base-transitive groups, the Tutte polynomial can be derived from the cycle index but not vice versa. I propose a polynomial for IBIS groups which generalises both Tutte polynomial and cycle index.

Metacyclic Invariants of Knots and Links

1970 ◽  
Vol 22 (2) ◽  
pp. 193-201 ◽  
Author(s):  
R. H. Fox
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Equivalence Class ◽  
Base Point ◽  
Covering Space ◽  
Transitive Permutation Group ◽  
Oriented Link ◽  
Inner Automorphism ◽  
Knots And Links

To each representation ρ on a transitive permutation group P of the group G = π(S – k) of an (ordered and oriented) link k = k1 ∪ k2 ∪ … ∪ kμ in the oriented 3-sphere S there is associated an oriented open 3-manifold M = Mρ(k), the covering space of S – k that belongs to ρ. The points 01, 02, … that lie over the base point o may be indexed in such a way that the elements g of G into which the paths from oi to oj project are represented by the permutations gρ of the form , and this property characterizes M. Of course M does not depend on the actual indices assigned to the points o1, o2, … but only on the equivalence class of ρ, where two representations ρ of G onto P and ρ′ of G onto P′ are equivalent when there is an inner automorphism θ of some symmetric group in which both P and P′ are contained which is such that ρ′ = θρ.

The Number of -Coloured Graphs

1968 ◽  
Vol 20 ◽  
pp. 1344-1352
Author(s):  
David A. Klarner
Keyword(s):  
Permutation Group ◽  
Cycle Index ◽  
Relevant Group ◽  
De Bruijn

In this paper we describe an algorithm for finding the number of non-isomorphic -coloured graphs with n nodes and e edges. We use Pόlya's fundamental enumeration theorem (in a form similar to that given by de Bruijn (see 1)) which reduces the problem to finding the cycle index for a certain permutation group. Harary (3) followed this same program for bi-coloured graphs, but failed to find the cycle index of the relevant group for general -coloured graphs.

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