Counting Polynomials and Related Indices by Edge Cutting Procedures
Mircea Vasile Diudea · 2010
A topological index is a numeric quantity derived from the structure of a graph G(V,E) which is invariant up to automorphisms of the considered graph. One of the most famous topological indices is the Wiener index W(G); it equals the sum of distances between all unordered pairs of vertices of G. A related number is the Szeged index SZ(G), which is the sum of all products of non-equidistant, proximal vertices nu(e), nv(e) with respect to the two ends of any edge e=(u,v) in G. Third is the Cluj index CJeS(G), calculated from the first derivative of CJe(x) polynomial. A forth index, called Cluj-Ilmenau CI(G), is calculated from the first and second derivatives of the Omega () x polynomial, which counts the opposite edge strips in G. All these indices and related polynomials are derived here by edge cutting procedures in some bipartite graphs and/or partial cubes. A clear relatedness among these descriptors was established and exemplified. Their use in correlating various physico-chemical or biological properties with the molecular structure have been extensively proven.