M. Klin, Ch. Rucker, G. Rucker, G. Tinhofer, "Algebraic combinatorics in mathematical chemistry I. Permutation groups and coherent algebras", Technische Universitat Munchen, TUM-M9510, 1995. 8  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(R ) 1; m ) can be computed in time O( m n)n 2 log n) 4 Final remarks In recent years, the importance and need of a fast implementation of the Weisfeiler Leman algorithm became more and more evident. In particular, for various problems occurring in mathematical chemistry (see e.g. [3, 12, 16]) the WL stabilization is now a well established tool. It is applied for the purpose of nding symmetries in molecular graphs and structures. We have developed an ecient implementation. A computer program is presented in [2] In that paper, we also showed how to modify the algorithm such that ....

M. Klin, Ch. Rucker, G. Rucker, G. Tinhofer, "Algebraic combinatorics in mathematical chemistry I. Permutation groups and coherent algebras", Technische Universitat Munchen, TUM-M9510, 1995. 8

....for instance in finite element computations. For recent surveys on graph Laplacians see [128, 122, 28, 129] 3.1.2. Coherent Algebras. An alternative, maybe even more appealing starting point is to consider the coherent algebra associated with the configuration space graph or transition operator [105, 183]. COMBINATORIAL LANDSCAPES 7 A set of complex matrices that is closed under (i) scalar multiplication with complex numbers, ii) component wise addition, iii) ordinary matrix multiplication, iv) component wise multiplication, and (v) transposition is called a coherent algebra or cellular ....

....centralizer algebra is coherent we have ##M## # VC (Aut[M] V ) 3. 2) Equality hold if and only if there is a permutation group that has ##M## as its centralizer algebra [106] The coherent algebra ##M## can therefore be regarded as a combinatorial approximation of the centralizer algebra [45, 105]. This is of particular importance in the graph case: given the adjacency matrix A of #, there is polynomial time algorithm that determines the coherent algebra W(#) ##A##, see [206, 9, 8] It is straightforward to check that the degree matrix D, and hence also the transition operator T = AD ....

M. Klin, C. R ucker, G. R ucker, and G. Tinhofer, Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algrebras, MATCH, 40 (1999), pp. 7--138.

....of the algorithm. 2 1 Introduction Coherent algebras have been introduced and studied by B. J. Weisfeiler and A. A. Lehman in [21] A combinatorially equivalent notion is the one of coherent configurations (see Higman [11] A short history of both of these and some related notions are given in [14]. Nowadays, coherent algebras are well studied objects in algebraic graph theory and have many applications in various areas of graph theory. Recently, results on recombination spaces [17] and recognition of circulant graphs [15] have been obtained by using coherent algebras. In addition, coherent ....

M. Klin, C. R ucker, G. R ucker, and G. Tinhofer, Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algebras., Tech. Rep. TUM-M9510, Technische Universitat Munchen, 1995.

.... the centralizer algebra is coherent we have hhMii V C (Aut[M] V ) 18) Equality hold if and only if there is a permutation group that has hhMii as its centralizer algebra [87] The coherent algebra hhMii can therefore be regarded as a combinatorial approximation of the centralizer algebra [34, 85]. This is of particular importance in the graph case: given the adjacency matrix A of Gamma, there is polynomial time algorithm that determines the coherent algebra W( Gamma) hhAii, see [157, 7, 6] Let R = fR (1) R (r) g be the standard basis of a coherent algebra W. We have R ....

M. Klin, C. Rucker, G. Rucker, and G. Tinhofer. Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algrebras. Technical Report TUM-M9510, TU Munchen, Garching, Germany, 1997.

....been built up around them in the literature. Today, the notion of coherent algebra and the equivalent notion coherent configuration (see [27] are among the main tools of algebraic combinatorics. A friendly introduction to coherent algebras taking into account the interests of chemists is given in [31], while the paper [13] is written for mathematicians and covers the most important theoretical aspects. We list here some properties of coherent algebras, relevant for the analysis which follows. For proofs see [26] i) Every coherent algebra possesses a unique linear basis A 1 , A s ....

....equals the number of connected components. For surveys on graph Laplacians see e.g. 36, 37] and the book [10] In order to show #,T # ##### it su#ces to verify that D # #####, which follows from the fact that the degree partition is coarser than the cell partition for any graph, see e.g. [31]. If # is regular, as is the case with the Robinson graphs, then A, T and # have the same eigenvectors since D is a multiple of the identity matrix. Let # k be an orthonormal basis of # with associated eigenvalues # k . The decomposition f = # k a k # k (14) is sometimes called a ....

M. Klin, C. Rucker, G. Rucker, and G. Tinhofer. Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algrebras. MATCH, 40:7--138, 1999.

....and of the implementation of Babel s algorithm. Finally, in Section 9, extended testing of our program on a large number of examples is documented in order to demonstrate its capability. We conclude with a discussion in Section 10. This work is the second contribution in a series of papers [KliRRT95], FurKT] concerning different aspects of algebraic combinatorics with emphasis on applications in mathematical chemistry. The series introduces the basic concepts of algebraic combinatorics and presents some of the main features and tools for perception of symmetry properties of combinatorial ....

....and presents some of the main features and tools for perception of symmetry properties of combinatorial objects. Those readers who are not familiar with mathematical standard definitions and notations such as matrix, group, basis, equivalence class, etc. are referred to the first paper [KliRRT95] in this series. However, we tried to make this work as selfcontained as possible and hope that it should be understandable for readers with a rather limited knowledge of mathematics. 5 2 Preliminaries An undirected graph is a pair Gamma = Omega ; E) consisting of finite sets Omega and E, ....

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Klin M., Rucker Ch., Rucker G., Tinhofer G.: Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algebras. Technical report TUM-M9510, Technical University of Munich, 1995.

....the implementation of Babel s algorithm. Finally, in Section 9, extended testing of our program on a large number of examples is documented in order to demonstrate its capability. We conclude with a discussion in Section 10. This work is the second contribution in a series of papers [KliRRT95] [FurKT] concerning different aspects of algebraic combinatorics with emphasis on applications in mathematical chemistry. The series introduces the basic concepts of algebraic combinatorics and presents some of the main features and tools for perception of symmetry properties of combinatorial objects. ....

....4 we obtain stronger algorithms, however at the price of a considerably higher complexity. One of the first attempts of a program implementation of stabilization of depth t 4 for purely chemical goals and on a rather naive level was done in [RueR91] It will be demonstrated in the next paper [FurKT] of this series that, in contrast to first expectations, stabilization of depth t with some t 4 is also not sufficient to rigorously settle the automorphism partitioning problem. It turns out (see [Fur87] CaiFI92] that for any fixed value of t there exist graphs with the property that the ....

Furer M., Klin M., Tinhofer G.: Algebraic combinatorics in mathematical chemistry. Methods and algorithms. III. Stabilization of graphs. Work in progress.

....and edges into the orbits and 2 orbits of the graph under consideration. 1.6 This paper is the third part of a series of publications dealing with the interrelations between algebraic combinatorics and mathematical chemistry. Formally it is selfcontained, nevertheless acquaintance with part I [KliRRT95] and part II [BabCKP97] certainly will help the reader to better understand the whole area of investigations, main concepts and motivations. 1.7 The paper consists of nine sections. All definitions related to isomorphisms and automorphisms of colored graphs together with some striking examples and ....

....are considered in Section 8. We also mention briefly the limitations of such approaches. In Section 9 we give some additional remarks and some conclusions are drawn from the material in the foregoing sections. 2 Isomorphisms and automorphisms of colored graphs 2. 1 We adopt the notation used in [KliRRT95]. Here we repeat the most important definitions concerning graphs and add some additional ones which are used in this paper. A directed graph Gamma = V; R) consists of a finite set V of vertices and a binary relation R, i.e. a subset R V Theta V: The size (or, alternatively, the order) of a ....

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M. Klin, C. Rucker, G. Rucker, G. Tinhofer, Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation Groups and coherent (cellular) algebras. Technical Report, Technische Universitat Munchen, TUM-M9510 (1995).

....such as coherent configurations [Hi70] relation algebras [Po93] block designs [BN39] association schemes [BM59] De73] strongly regular graphs [Bo63] Hu75] and many others. For an introduction into the topic of coherent algebras and a review of important applications we refer to the paper [KRRT95]. Our main motivation to study coherent algebras stems from their relation to two notoriously hard graph theoretical problems, namely the problems to decide whether two graphs are isomorphic or not and to find the automorphism partition of a graph. More formally, two graphs G = X; R) and G = ....

M. Klin, C. Rucker, G. Rucker, G. Tinhofer, Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation Groups and coherent (cellular) algebras. Technical Report, Technische Universitat Munchen, TUM-M9510 (1995).

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