An Application of the permanent-determinant method: computing the Z-index of trees

Transcription

1 Department of Mathematics wtrnumber An Application of the permanent-determinant method: computing the Z-index of trees Daryl Deford March 2013 Postal address: Department of Mathematics, Washington State University, Pullman, WA Voice: Fax: URL:

2 AN APPLICATION OF THE PERMANENT DETERMINANT METHOD: COMPUTING THE Z INDEX OF TREES DARYL DEFORD Abstract. Topological invariants of molecules relate the structure of their underlying graph to physical properties of the chemical. One of the first topological invariants to be considered in computational chemistry was the Z index, introduced by Hosoya in 1971, which is equivalent to the total number of matchings in the underlying graph [7]. In this paper we show that the Z index of any tree can be computed as the determinant of a specifically constructed matrix, using techniques from graph theory, linear algebra, and combinatorics. In addition, we demonstrate how this method can be extended to weighted trees and more general chemical structures. 1. Introduction Inthispaperweprovethatthenumberofmatchingsofanytreecanbecomputed as the determinant of an n n matrix constructed from the adjacency matrix of the tree. This value is equivalent to the Z index of the tree, when the graph is chemically motivated Background. Given a chemical molecule, a graph can be constructed that represents the most basic structure of the molecule by placing a vertex in the graph for each atom and an edge between two vertices if their respective atoms share a bond. Some properties of the molecule can be determined from the attributes of its respective graph. These relations between graph metrics and physical properties are called topological invariants or indices because they remain unchanged under isomorphisms [12]. The Z index is a frequently studied topological invariant, defined by Hosoya in 1971, that correlates with boiling points of linear alkanes and other physical properties [7]. To define the Z index of a chemical structure, C, we must first define a function, p(c, k), that counts the number of subgraphs with k connected components, each isomorphic to K 2, on the graph corresponding to C [15]. Then, the Z index of a molecule with n atoms can be represented as Z(C) = n/2 k=0 similarly a Z polynomial may be defined as P z (C,x) = n/2 k=0 p(c,k), p(c,k)x k. Date: March 23, Mathematics Subject Classification. Primary 92E10; Secondary 05C05. Key words and phrases. Matrix Permanent; Hosoya Index; Cycle Cover; Matchings. 1

3 2 DARYL DEFORD Notice that these are respectively equivalent to the total number of matchings and matching polynomial of the corresponding graph [1, 11]. Thus, the Z index of a chemical is equivalent to the total number of matchings of the graph or P z (1). In general, computing the matching polynomial or total number of matchings of an arbitrary graph is #P complete and computationally infeasible for even moderately sized graphs[11]. Although the number of perfect matchings can be calculated in polynomial time for planar graphs, the same is not true of the total number of matchings or matching polynomial. A tree is defined to be a connected graph with no cycles [3]. It is easy to see that all trees are both bipartite and planar. As structures, trees are particularly important in computational chemistry, since many families of molecules, such as the alkanes, can be represented as trees [2, 3]. Trees are also commonly used in computer science to model a variety of objects, from data structures to algorithms and even hardware design [4] Related Work. Previous work by Gutman, et. al, showed that the Z index of a tree can be computed as a product of functions of the eigenvalues of an adjacency matrix [5]. Later, Ahmadi and Dastkhezr proved that the Z index of a tree is also equal to the sum of the absolute values of the coefficients of the characteristic polynomial of the adjacency matrix [1]. Several papers have also been published giving upper and lower bounds of the magnitudes of Z indices for graphs with fixed numbers of vertices, for example [13]. In this paper we use the theory of matrix permanents to show that if A(C) is the graph adjacency matrix of a molecule with n atoms, whose corresponding graph is a tree, that the determinant of A(C)i+I n is equal to the Z index of the chemical. Similar results to those presented in this paper have been motivated by considering continued fraction tree expressions. A tri-diagonal matrix with similar properties was originally defined by Hosoya and Gutman in [8]. Hudleson then gives a generalization of their result, showing that the determinant of a properly constructed, signed B matrix associated with a tree is also equal to the Z index of that tree [9]. We will also prove similar results for weighted trees and other structures that commonly occur in computational chemistry Matrix Permanents and Cycle Covers. Given a graph, G = (V, E), representing a chemical structure, C, we will represent the adjacency matrix of G as A(G). The n n identity matrix will be denoted I n. The permanent of a matrix, M, with elements M u,v, written per(m), is defined as the unsigned sum of all the permutations of the matrix [6, 14]. Thus, per(m) = n M i,π(i), π S n i=1 is a symbolic representation of the matrix permanent. Permanents, even of 0 1 matrices, are in the complexity class #P and are thus very inefficient to compute directly[11,16]. Althoughthedefinitionofthepermanentisveryclosetothatofthe determinant, the permanent shares very few of the determinants useful properties or relations to eigenvalues. However, interchanging rows or columns of the matrix does not affect the value of the permanent of that matrix [14]. A cycle cover, or linear subgraph, is a subset of the edges of a directed graph (digraph) that covers all vertices of the digraph in such a way that each vertex has in degree and out degree equal to one. Matrix permanents occasionally appear in

4 Z INDEX OF TREES 3 (a) T (b) T (c) T l Figure 1. A Tree and its Corresponding Digraphs combinatorial problems, because the permanent of the adjacency matrix of a digraph is equal to the number of cycle covers of that digraph [6]. Kuperberg s paper includes a survey and description of the ways that permanents are used in combinatorics, as well as describing several applications of the method to computational chemistry [10]. Similarly, the permanent of a bi adjacency matrix of a bipartite graph counts the number of perfect matchings in the graph [6] Digraphs. Let G = (V,E) be a graph, with adjacency matrix A(G). We will constructadigraph, G, fromgbyreplacingeachsimpleedgeingwithtwodirected edges, oneineachorientation. NotethatbydefinitionwehaveA( G) = A(G), since for a digraph, D, A(D) u,v = 1 implies that there is a directed edge from vertex u to vertex v. Next, we add a self loop to each vertex in G to form G l. This is equivalent to adding the appropriate identity matrix to A( G), thus we have A( G l ) = A( G)+I n. Figure 1 shows a tree and its associated digraphs. 2. Main Results In this section we will prove that the Z index of any tree can be computed by taking the determinant of a particular modification of the adjacency matrix. We will proceed by providing a bijection between directed cycle covers and matchings and then construct a matrix whose determinant is equal to this value Trees. Let T = (V,E) be an n vertex tree with adjacency matrix A(T). Then the following result follows directly from the definitions given above. Lemma 1. The number of cycle covers of T l is equal to per(a(t)+i n ). Proof. The adjacency matrix of T l is equal to (A(T)+I n ) by construction. Since per(a( T l ))isequaltothenumberofcyclecoverson T l, per(a( T l )) = per((a(t)+ I n )), which implies the lemma.

5 4 DARYL DEFORD Now we can prove the relationship between cycle covers and matchings which will allow us to compute the Z index of a given tree. Theorem 1. The number of cycle covers on T l is equal to the number of matchings on T. Proof. It suffices to provide a bijection between the cycle covers on T l and the matchings of T. Since T is a tree it has no cycles, and thus T l only contains 1 cycles, consisting of self loops, and 2 cycles, between vertices adjacent in T. To construct a unique matching on T from a cycle cover of T l, place an edge in the matching between two vertices, u and v, if and only if u and v lie on a 2 cycle in the cycle cover. Similarly, to determine a unique cycle cover from a matching on T, place a 2 cycle in the cycle cover for each edge in the mapping, and place each remaining vertex in T l on a 1 cycle. Since this relation is a bijection it must be that the set of cycle covers on T l has cardinality equal to the number of matchings on T and this proof is complete. Remembering that the Z index of T is equal to the number of matchings on T, the obvious corollary to this theorem allows us to calculate Z(T) as a matrix permanent. Corollary 1. The Z index of T is equal to per(a(t)+i n ). Proof. This result follows directly from the application of Lemma 1 to the result of Theorem 1. Since calculating the permanent of a matrix is computationally difficult, we wish construct a matrix whose determinant is equal to per(a(t)+i n ), because we can calculate determinants in polynomial time [4]. The sign of a permutation in the determinant can be computed as the parity of the number of even cycles in the permutation. A permutation with an even number of even cycles is positive, while a permutation with an odd number of even cycles is negative. Thus, we want to weight the entries of A(T)+I n so that each odd permutation has a weight of 1 to cancel out the sign of the permutation. We now prove that giving each directed edge between two distinct vertices a weight of 1 = i is sufficient. Theorem 2. For any tree, T, on n vertices, per(a(t)+i n ) = det(a(t)i+i n ). Proof. We have already seen that per(a(t)+i n ) counts the number of cycle covers of T l andthatthesecyclecoverscontainonly2 cyclesand1 cycles. Each2 cycle corresponds to a product of 2 non-main diagonal entries of A(T)+I n. Thus, giving each of these entries a weight of i gives each 2 cycle, and thus each even cycle, a weight of i 2 = 1. Since each 2 cycle in A(T)i+I n has weight negative one, each cycle cover with an odd number of even cycles has weight ( 1) 2k+1 = 1 and each cycle cover with an even number of even cycles has weight ( 1) 2k = 1. This further implies that for each cycle cover (permutation) the sign of the cycle cover (permutation) is equal

6 Z INDEX OF TREES 5 to the weight of the cycle cover (permutation). Thus, we have det(a(t)i+i n ) = π S n sgn(π) n i=1 (A(T)i+I n) i,π(i) = π S n sgn(π)sgn(π) n i=1 (A(T)+I n) i,π(i) n = π S n i=1 (A(T)+I n) i,π(i) = per(a(t)+i n ) and the proof is complete. Corollary 2. The Z index of any tree, T, on n vertices, is equal to det(a(t)i+i n ). Proof. Again this follows directly from the preceding results. Thus, we have proved that the Z index of any tree can be computed by multiplying its adjacency matrix by i, adding ones along the main diagonal and taking a determinant. Hence, the Z index of any chemical compound whose structural graphs is a tree can be found by simple matrix operations in polynomial time Example. Consider the tree shown in Figure 1. Its adjacency matrix and modified adjacency matrix are shown below, and it is trivial to verify that: Z(T) = per(a(t)) = det(a(t)i+i 9 ) = 33. A(T) = i i 1 i i i i i A(T)i+I 9 = 0 i i i 1 i i i i 0 1 i i 1

7 6 DARYL DEFORD 2.3. Generalizations. Here we consider some extensions to the previous results Weighted Trees. Given a weighted tree, T the sum of the weights of its matchings may be computed in a similar fashion. To form a matrix whose determinant is equal to this value, use a weighted adjacency matrix A w (T) where the weight of each edge is placed in the upper triangular portion of the adjacency matrix. Corollary 3. For a weighted tree T, the sum of the weights of all matchings of T is equal to det(a w (T)i+I n ). Proof. Since T is a tree, and thus T l contains only 1 cycles and 2 cycles, we have from Theorem 2 that det(a w (T)i+I n ) = per(a w (T)+I n ). The matrix permanent on the left hand side sums the product of the weights of each cycle for each cycle cover. Each 1 cycle has a weight of 1 from the main diagonal, and each 2 cycle gains its weight from the upper triangular entry in the cycle multiplied by the corresponding 1 in the lower triangular portion. Making use of the bijection provided in Theorem 1 we see that each weighted cycle cover in the permanent may be paired directly with a matching of T the same weight and this proof is complete Graphs With One Cycle. Not all chemical structures can be represented as trees. Here we show that the Z index of a graph with exactly one cycle can be computed by taking two determinants. Theorem 3. Let G be a graph on n vertices with exactly one k cycle, C k, with k n. The Z index of G is equal to if k is odd, and if k is even. det(a(g)i+i n ) 2i k (det(a(g C k )i+i n k )) det(a(g)i+i n )+2i k (det(a(g C k )i+i n k )) Proof. The given determinant is computing the signed weights of all cycle covers of G. These cycle covers all correspond to matchings by the bijection of Theorem 1 except for those that contain C k in one of two orientations. When C k is part of a cycle cover it contributes weight i k to the product, since it contains exactly k edges of weight i. The rest of the cycle cover consists of a matching on the remaining vertices of G C k. We know that G C k has no cycles, although it may not be connected, thus by Theorem 2 the matchings of G C k are counted by det(a(g C k )i+i n k ). Thus, det(a(g)i+i n ) counts 2 det(a(g C k )i+i n k ) extra cycle covers each of weight i k where C k is included in the cycle cover, once in each orientation. When C k is an odd cycle it does not contribute to the sign of the overall permutation, but when C k is an even cycle it changes the parity of the permutation. Thus, when C k is an even cycle the over counted permutations, computed as det(a(g C k )i + I n k ), must be multiplied by 1. Subtracting these two quantities gives the desired result and the proof is complete.

8 Z INDEX OF TREES 7 3. Examples We conclude this paper by presenting some chemically motivated examples of the methods detailed above. Figure 2. C 2 H 7 NO Figure 3. C 6 H 18 N 2 O 2

9 8 DARYL DEFORD Figure 4. C 8 H 6 ClNO aminoethanol. The chemical composition of 2-aminoethanol is C 2 H 7 NO and a structure isomorphic to its underlying graph is shown in Figure 2. The adjacency matrix and modified adjacency matrix are shown below, and it is easy to verify that Z(C 2 H 7 NO) = per(a(c 2 H 7 NO)) = det(a(c 2 H 7 NO)i+I 11 ) = 76. A(C 2 H 7 NO) = A(C 2 H 7 NO)+I 11 = i i i i 1 i i 1 i i i i i i i i i i i i i i 1

10 Z INDEX OF TREES hexane-2, 5-dione oxime. The chemical formula for hexane-2, 5-dione oxime is C 6 H 12 N 2 O 2, and its structural graph is isomorphic to a tree with two edges of weight two. The first matrix shows the standard adjacency matrix of the graph while the second matrix shows the modified, weighted digraph adjacency matrix. Again, it can be verified that Z(C 6 H 18 N 2 O 2 ) = per(a(c 6 H 12 N 2 O 2 )) = det(a w (C 6 H 12 N 2 O 2 )i+i 18 ) = A w (C 6 H 12 N 2 O 2 ) = A w (C 6 H 12 N 2 O 2 )i+i 18 = 1 i i 1 i i 1 2i i 1 i i i 1 i i i i i i i i i 1 i i 1 2i i i 1 i i i i i i i i i 1 i i 1

11 10 DARYL DEFORD 3.3. benzene, 1 chloro 4 (2-nitroethanol). The chemical formula for benzene, 1 chloro 4 (2-nitroethanol) is C 8 H 6 ClNO 2, and as can be seen from Figure 4 the structure of the molecule contains a benzene ring, which is a 6 cycle. The first two matrices below represent the entire molecule, while the second two matrices describe the vertices left after the 6 cycle is removed. A calculation motivated by Theorem 3, similar to those above gives our final result: Z(C 8 H 6 ClNO 2 ) = det(a(c 8 H 6 ClNO 2 )i+i 12 )+2i k (det(a(c 8 H 6 ClNO 2 C 6 )i+i 6 )) = 270. A(C 8 H 6 ClNO 2 ) = A(C 8 H 6 ClNO 2 )i+i 12 = 1 i i 1 i i i 1 i i 1 i i 1 i 0 i i 1 i i i i i i 1 i i 1 i i 1 i i 1 A(C 8 H 6 ClNO 2 C 6 ) =

12 A(C 8 H 6 ClNO 2 C 6 )i+i 6 = Z INDEX OF TREES 11 Acknowledgements i i 1 i i 1 i i i i 0 1 The author would like to thank Dr. William Webb and Dr. Matthew Hudelson for their insight and assistance. This work was supported by a grant from the Washington State University College of Sciences. References [1] M. Ahmadi and H. Dastkhzer: On the Hosoya index of trees, Journal of Optoelectronics and Advanced Materials, 13(9), (2011), [2] A. Balaban ed.: Chemical Applications of Graph Theory, Academic Press, London, 1976 [3] G. Chartrand, L. Lesniak, and P. Zhang: Graphs & Digraphs Fifth Edition, CRC Press, Boca Raton, [4] T. Cormen, C. Leiserson, R. Rivest, and C. Stein: Introduction To Algorithms Third Edition, MIT Press, Cambridge, [5] I. Gutman, Z. Markovic, S.A. Markovic: A simple method for the approximate calculation of Hosoya s index, Chemical Physical Letters, 132(2), 1987, [6] F. Harary: Determinants, Permanents and Bipartite Graphs, Mathematics Magazine, 42(3),(1969), [7] H. Hosoya: The Topological Index Z Before and After 1971, Internet Electron. J. Mol. Des. 1, (2002), [8] H. Hosoya and I. Gutman: Kekulè Structures of Hexagonal Chains Some Unusual Connections, J. Math. Chem. 44, (2008), [9] M. Hudelson: Vertex Topological Indices and Tree Expressions, Generalizations of Continued Fractions, J. Math. Chem. 47, (2010), [10] G. Kuperberg: An Exploration of the Permanents-Determinant Method, Electronic Journal of Combinatorics, 5, (1998), R46: [11] P. Lundow: Computation of matching polynomials and the number of 1-factors in polygraphs, Research Reports Umeå, 12, (1996) [12] D. Rouvray ed.: Computational Chemical Graph Theory, Nova Science Publishers, New York, 1990 [13] R. Tichy and S Wagner: Extremal Problems for Topological Indices in Combinatorial Chemistry, Journal of Computational Biology, 12(7), (2005), [14] N. Trinajstic: Chemical Graph Theory Volume I, CRC Press, Boca Raton, 1983 [15] N. Trinajstic: Chemical Graph Theory Volume II, CRC Press, Boca Raton, 1983 [16] L. Valiant: The Complexity of Computing the Permanent, Theoretical Computer Science, 8(2), (1979), Department of Mathematics, Washington State University, Pullman, Wa address: daryl.deford@ .wsu.edu