Solving Eigenvalues Applications from the Point of Theory of Graphs


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1 Solving Eigenvalues Applications from the Point of Theory of Graphs Soňa PAVLÍKOVÁ Faculty of Mechatronics, AD University of Trenčín Abstract Eigenvalues of linear mappings are applied in many applications in physics, electrical engineering and in other areas of science, especially in fields of military technologies. The dimer problem will be described in the paper and possible ways to resolve this problem in terms of graphs spectrums will be presented. There are many methods for finding eigenvalues of matrixes, either by decomposition of a matrix to canonical forms or numerical methods. The solution of this problem will be explained from the viewpoint of perspective graph theory. Each matrix can be represented as a labeled graph and eigenvalues of the matrix then correspond to the graph spectrum. In case, where a linear mapping matrix is similar to an unsigned integer matrix, other possibilities of further examination of eigenvalues in terms of graph theory are available. Also the necessary and sufficient condition, when a square matrix is diagonally similar to unsigned matrix, will be introduced. Since the possible range of technical problems coverage is much more over the size of this paper, the concrete application for technical problems will be introduced in the next paper of the author. Key words: dimer problem, graph, spectrum of graphs, eigenvalues of matrices Introduction The solution of technical problems is nowadays as close to orthodox mathematics as have never been before. For example, a lot of designers and scientists are using finite elements or other numerical methods with only basic (if any) knowledge of the mathematical background of numerical mathematics and partial differential equations. The previously mentioned example covers the most popular and widespread conjunction of mathematics to designers everydays life. Nevertheless, there are other (sometimes strong, sometimes week) influences of mathematics one of a few explored area is the graph theory. Before the graph theory began wellknown, the technical problems have been rewritten according to the given (and usually very complicated) rules into the form of matrixes. One of the most important indicators is the eigenvalues of the matrixes. The consecutive analysis is always based on theirs evaluation. Any technical problem (could be based on electrotechnical engineering, mechanics, micro and nano tech, civil engineering or chemistry) can be more or less easily transformed into the structure of oriented graphs. These can be used either for storing the data in less space and time consuming way; or, and this is the way of this paper) for analysis and testing the structure properties. When transforming the technical problem into the form of oriented
2 graphs, the eigenvalues of matrixes can be obtained in the simpler way and this will spare designer s time and performance of CPU. The important technique used in this kind of analysis is called linear mapping. Eigenvalues and eigenvectors of linear mappings are widely applied in solving of many applications in electrical engineering, chemistry or economy. Linear mappings can be characterized as a mapping between two vector spaces that retain the coefficients of linear combination of vectors in two given spaces. Every linear mapping has its own matrix representation. There are many ways how to solve the problem of eigenvalues. Methods of linear algebra use the similarity of matrixes. It is well known that similar matrixes have the same eigenvalues. Every square matrix is similar to the matrix in rational canonical form. Eigenvalues of a matrix can be solved by numerical methods. The eingenvalue of A are the roots of the characteristic polynomial of A. Once the characteristic polynomial is known, the roots can be solved by using subroutines for solving the roots of polynomials. However a polynomial may be ill  conditioned in the sense that the small changes in coefficients may cause large changes in one or more roots. Hence this procedure of computing the eigenvalues may change a well  conditioned problem into an ill conditioned problem and should be avoided. Therefore, it is appropriate to look for other ways to solve this problem. The problem of eigenvalues can be also solved through the graph theory, which brings up many nontraditional solutions. Background of the Problem 1,..., n Let G is a finite nonoriented graph without loops with a vertex set V G v v and an edge set of graph G is E G (multiple edges are allowed). Adjacency matrix A a ij a square matrix of n x n, whose element vertex v i with the vertex a ij is equal to the number of edges connecting the v j. Obviously, the graph G is explicitly determined (but isomorphism) by matrix A, but vice versa it is not true  matrix A depends on the labeling of the vertixes. Therefore, we can assign the matrix class M=M(A ) to the graph G, where two matrixes A and B belong in this class if and only if, where there is such a permutation matrix 1 P that B P AP. Important invariant of the class M is the characteristic polynomial P I A, where i, i 1,2,..., n is the root of the equation PG ( ) 0 and n. For analyzing eigenvalues of a matrix using assigned graph, it is important that the matrix is similar to nonnegative matrix. In the results we will present the condition, when the matrix is similar to nonnegative matrix. Bipartite graphs play an important role in the following motivation example. Graph G will be called bipartite if it does not contain a circle of an odd length. Vertex set of bipartite graph can be G det( ), respectively Sp( G) 1, 2,..., n divided into two disjoint subsets R and C, that R C V( G) and no two vertixes from the set R, respectively C are connected by an edge. F factor of graph G is a such subgraph of graph G, that V (F) = V (G) and E( F) E( G). The Dimer Problem The spectra of graphs, or the spectra of certain matrices which are closely related to the adjacency matrices appear in a number of problems in statistical physics [1], [2], [3]. We shall describe so called dimer problem.
3 The dimer problem is related to the investigation of the thermodynamic properties of a system of diatomic molecules ( dimers ) adsorbed on the surface of a crystal. The most favorable points for the absorption of atoms on such surface form a twodimensional lattice, and a dimer can occupy two neighboring points. It is necessary to count all ways in which dimers can be arranged on the lattice without overlapping each other, so that every lattice point is occupied. The dimer problem on a square lattice is equivalent to the problem of enumerating all 1 2 ways in which a chessboard of dimension n x n (n being even) can be covered by 2 n dominoes, so that each domino covers two adjacent squares of the chessboard and that all squares are so covered. A graph can be associated with a given adsorption surface. The vertices of the graph represent the points which are the most favorable for adsorption. Two vertices are adjacent if and only if the corresponding points can be occupied by a dimer. In this manner an arrangement of dimers on the surface determines a factor in the corresponding graph, and vice versa. Thus, the dimer problem is reduced to the task of determining the number of factors in a graph. Such assigned graphs are bipartite and fall into special classes of graphs, which we mark A. The reader will find a detailed description of the class A of graphs for example in [4]. The basic Theorem [4] can be stated as:. Let G A.. The number of factors of G is equal to the product of all non negative eigenvalues of G. Results and Summary Theorem 1 can be stated as: Let A is a square matrix of order n. Then A is diagonally similar to unsigned matrix if and only if there is a decomposition of the index set 1,2,...,n for nonempty sets of and with the property that A and A A and A are nonpositive matrixes. Lema 1: are nonnegative matrixes and Proof of the Theorem 1 is a consequence from the following statements: Let A a ij and B b ij same zero points, i.e. aij 0 if and only if bij 0. Lema 2: every i, j: Let A a ij and B b ij are diagonally similar matrixes. Then A and B have the are square matrixes, with the following properties for
4 (i) if bij 0 then aij 0 (ii) if aij c, where c is a real number different from zero, then bij aij. Then, diagonal similarity of matrix B with unsigned matrix implies a diagonal similarity of matrix A with nonnegative matrix. Lema 3: Let A be a square matrix of order n. Then A is diagonally similar with some unsigned matrix if and only if A is (0, 1, 1)  diagonally similar with some nonnegative matrix. The following concequence can be derived from Theorem 1. Let A be a square matrix of order n. Then A is diagonally similar with some nonnegative matrix if and only if there is such permutation matrix P that AA PAP AA 21 2 where A 11 and A 22 are nonnegative square matrixes and matrixes A 12 and A 21 are nonpositive. Conclusions The scope of this paper shows options available when using the graph spectrums. Spectra of graphs have appeared frequently in mathematical literature since the fundamental papers [5]. Even earlier, starting from the thesis of [6] in 1931, theoretical chemists were interested in graph spectra, although they used different terminology. The possibilities of solving eigenvalues using graph theory have been outlined in the paper. Problems of graphs spectra open up many so far unexplored possibilities in applications and give an unusual insight into many problems of physics, chemistry, electrical engineering etc. very commonly used to solve the problems of special and military technologies. The aim of this paper is to give the reader the necessary mathematical background of the presented objectives. The concrete technical applications will be introduced in the next paper. References 1. KASTELEYN, P. W. Graph Theory and theoretical Physics. London  New York : Academic Press, Graph theory and crystal physics, pp MONTROL, E. W. Applied Combinatorial Mathematics. London  New York  Sydney: Interscience, John Wiley &Sons, Latice statistics, pp PERCUS, J. K. Combinatorial Methods. Berlin  Heidelberg  New York : Springer  Verlag, CVETKOVIĆ, D. M.; DOOB, M.; SACHS, H. Spectra of Graphs : Theory and Application. Second edition. Berlin: VEB Deutscher Verlag der Wissenschaften, p. 5. SINOGOWITZ, U., COLLATZ, L. Spektren endlicher Grafen. Abh. Math. Sem. Univ. Hamburg. 1957, 21, pp HÜCKEL, E. Quantentheoretische Beiträge zum Benzolproblem. Z. Phys , 70, pp
5 7. MATEJDES, M. Aplikovaná matematika. Zvolen: MAT  CENTRUM, p. ISBN
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