scholarly journals Chromatic Classes of 2-Connected (n,n+4)-Graphs with Exactly Three Triangles and at Least Two Induced 4-Cycles

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
G. C. Lau ◽  
Y. H. Peng
Keyword(s):  
Chromatic Polynomial ◽  
Equivalence Classes ◽  
Chromatic Equivalence

For a graph G, let P(G,λ) be its chromatic polynomial. Two graphs G and H are chromatically equivalent, denoted G∼H, if P(G,λ)=P(H,λ). A graph G is chromatically unique if P(H,λ)=P(G,λ) implies that H≅G. In this paper, we determine all chromatic equivalence classes of 2-connected (n,n+4)-graphs with exactly three triangles and at least two induced 4-cycles. As a byproduct of these, we obtain various new families of χ-equivalent graphs and χ-unique graphs.

scholarly journals Chromatic equivalence classes of certain generalized polygon trees

1997 ◽  
Vol 172 (1-3) ◽  
pp. 103-114 ◽  
Author(s):  
Yee-Hock Peng ◽  
C.H.C. Little ◽  
K.L. Teo ◽  
H. Wang
Keyword(s):  
Equivalence Classes ◽  
Generalized Polygon ◽  
Chromatic Equivalence

scholarly journals Chromatic equivalence classes of certain cycles with edges

2001 ◽  
Vol 232 (1-3) ◽  
pp. 175-183 ◽  
Author(s):  
Behnaz Omoomi ◽  
Yee-Hock Peng
Keyword(s):  
Equivalence Classes ◽  
Chromatic Equivalence

scholarly journals Jones Representations of Thompson’s Group F Arising from Temperley–Lieb–Jones Algebras

2019 ◽  
Author(s):  
Valeriano Aiello ◽  
Arnaud Brothier ◽  
Roberto Conti
Keyword(s):  
Finite Type ◽  
Chromatic Polynomial ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Parametric Family ◽  
Matrix Coefficients ◽  
Thompson’S Group ◽  
Planar Algebra ◽  
Index Value ◽  
Thompson's Group F

Abstract Following a procedure due to Jones, using suitably normalized elements in a Temperley–Lieb–Jones (planar) algebra, we introduce a 3-parametric family of unitary representations of the Thompson’s group $F$ equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behavior at infinity of their matrix coefficients, thus showing that these representations do not contain any finite-type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of $F$. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of $F$ indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the 1st non-trivial index value for which the corresponding subgroup is isomorphic to the Brown–Thompson’s group $F_3$, we show that when the index is large enough, this subgroup is always trivial.

scholarly journals MET: a Java package for fast molecule equivalence testing

2020 ◽  
Vol 12 (1) ◽  
Author(s):  
Jördis-Ann Schüler ◽  
Steffen Rechner ◽  
Matthias Müller-Hannemann
Keyword(s):  
Chemical Properties ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Equivalence Classes ◽  
Equivalence Testing ◽  
Equivalence Test ◽  
Graph Isomorphism Problem ◽  
2D Structure ◽  
Node Labels ◽  
Labelled Graphs

AbstractAn important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

Parallel and Distributed Processability of Objects

1989 ◽  
Vol 12 (3) ◽  
pp. 317-356
Author(s):  
David C. Rine
Keyword(s):  
Software Engineering ◽  
Software Design ◽  
Equivalence Classes ◽  
Minimum Amount ◽  
Software Components ◽  
Limited Extent ◽  
Design Environment ◽  
Distributed Software

Partitioning and allocating of software components are two important parts of software design in distributed software engineering. This paper presents two general algorithms that can, to a limited extent, be used as tools to assist in partitioning software components represented as objects in a distributed software design environment. One algorithm produces a partition (equivalence classes) of the objects, and a second algorithm allows a minimum amount of redundancy. Only binary relationships of actions (use or non-use) are considered in this paper.

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