Fast Partial-Differential Synthesis of the Matching Polynomial of C 72-100 James M. Salvador,* Adrian Hernandez, Adriana Beltran, Richard Duran, and Anthony Mactutis Department of Chemistry, University of Texas at El Paso, El Paso, Texas 79968 Received February 11, 1998 The Thesis algorithm uses partial-differential edge operators and a grammatical structure to generate and avoid expanding the Matching Polynomial. To run the algorithm efficiently, the vertexes of fullerene graphs C 60-100 were sorted into three-dimensional sectors. INTRODUCTION The Matching, 1 Reference, 2 or Acyclic 3 Polynomial is the counting polynomial of nonadjacent edges of a graph as a function of isolated vertexes. Its lowest order term is the number of perfect matchings or Kekule ´ structures of a given π-system. Hosoya’s Z index is the number of terms in a Matching Polynomial. The Z index is elegantly calculated through graph “retrosynthesis” and easily correlates the hydrogen suppressed graph of acyclic alkanes to their boiling point. The difference in the “occupied” roots of the Matching and Characteristic Polynomials provides a method of comparing the stability of different size π-systems and is called a Topological Resonance Energy (TRE). Because of recent interest in calculating the TRE of fullerenes up to C 70 , 4-8 we report the development of the Thesis algorithm to quickly calculate the Matching Polynomial of larger fullerenes using partial-differential edge operators. In matrix terms, for a graph G the Characteristic Polyno- mial, CP(G) or the determinant of its adjacency matrix with elements a i,j , is a listing of all walks across the matrix without stepping twice on the same row or column. The Matching Polynomial, MP(G), is a listing of all determinant walks that contain only matching steps a i,j and a j,i . MP(G) equals CP- (G) for acyclic graphs. MP(G) is a subset of CP(G) for cyclic graphs because the latter includes walks that explicitly list the Hamiltonian paths of the graph. Matching off-diagonal walks are permutations of the diagonal walk since the diagonal and off-diagonal positions of an adjacency matrix can correspond to the vertexes (V i ) and edges (e i,j ) of G, respectively, as shown in Figure 1. Rosenfeld and Gutman showed that these permutations are a product of partial-differential operators on the product of diagonal elements. 9 We use partial-differential edge opera- tors to develop Thesis as follows. METHODS MP(G) for the trivial graph of n vertexes is the product of the diagonal elements of the adjacency matrix, eq 1. An edge operator is a permutor of the diagonal walk to its matching off-diagonal elements, eq 2. MP(G) for a nontrivial graph is the product of edge operators on the trivial graph monomial as shown in eq 3. Partial-differential operators ensure that no row or column is repeated in a walk since the Figure 1. An adjacency matrix. Figure 2. A Mathematica program to generate MP(C 60 ). Figure 3. (((V 1 V 2 )(V 3 V 4 ))((V 5 V 6 )(V 7 V 8 ))). 1105 J. Chem. Inf. Comput. Sci. 1998, 38, 1105-1110 10.1021/ci9800155 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/18/1998