Fast Partial-Differential Synthesis of the Matching Polynomial of C_72-100

Abstract

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 C60-100 were sorted into three-dimensional sectors.

The topology of fullerenes

“…Balasubramanian gave polynomial coefficients c ( G , k ) for a number of fullerenes up to C 50 , 232 Babić determined the coefficients for C 70 , 216 and Salvador for C 60 and C 70 to C 100 . 233 The first few polynomial coefficients are known as they are independent of the isomers for a specific vertex count, that is, c ( G , 0) = 1, c ( G , 1) = − 3 N /2, c ( G , 2) = 3 N (3 N − 10)/8, c ( G , 3) = − (9 N 3 − 90 N 2 + 232 N )/16. 232 The last coefficient c ( G , N /2) is just the number of perfect matchings in the graph.…”

“…The computation of these coefficients soon becomes computationally intractable. 233 Babić et al found, however, a good correlation between E MP and E π values for fullerenes, and the E π value suffices to calculate the topological resonance energy approximately as (in units of β ), 222 …”

Symbolic calculation in chemistry: Selected examples

“…Salvador et al computed matching polynomials of fullerene graphs for C 60–100 by an algorithm that uses partial differential edge operators 259.…”

The cycle (circuit) polynomial of a graph with double and triple weights of edges and cycles

“…Although a repetitive action of matrix operators on the R. H. S.'s of (20) and (21) is already a true theoretical way to calculate respective polynomials φ − m (G; x) and φ + m (G; x), we want to derive special matrix operators that perform the double and triple action of the mentioned operators just for one application. LetĈ = [ĉ ij ] p i,j=1 be another matrix with an entryĉ ij = a ij ∂/∂x i , where a ij is an entry of the (weighted) adjacency matrix A.…”

“…Notice also that our previous approach [10,11] has already been implemented in symbolic computer codes and practically employed by Salvador et al [21] and Cash [22] for computing the matching and permanental polynomials of molecular graphs.…”