The Padmakar-Ivan (PI) index of a graph G is defined as PI (G) = Σ [n eu (e|G)+n ev (e|G)], where for edge e= (u,v) are n eu (e/G) is the number of edges of G lying closer to u than v, n ev (e/G) is the number of edges of G lying closer to v than u and summation goes over all edges of G. the PI index is a Wiener-Szeged-Like topological index developed very recently. In this paper we describe a method of computing PI index of benzenoid hydrocarbons (H) using orthogonal cuts. The method requires the finding of number of edges in the orthogonal cuts in a benzenoid system (H) and the edge number of H -a task significantly simpler than the calculation of PI index directly from its definition.
AbstractThe Padmakar-Ivan (PI) index of a graph G is defined as PI (G) = Σ [n eu (e|G)+n ev (e|G)], where for edge e=(u,v) are n eu (e|G) the number of edges of G lying closer to u than v, and n ev (e|G) is the number of edges of G lying closer to v than u and summation goes over all edges of G. The PI index is a Wiener-Szeged-like topological index developed very recently. In this paper we describe a method of computing PI index of benzenoid hydrocarbons (H) using orthogonal cuts. The method requires the finding of number of edges in the orthogonal cuts in a benzenoid system (H) and the edge number of H -a task significantly simpler than the calculation of PI index directly from its definition.
Let G be a finite bipartite plane graph. In a chemical context, a set of pairwise disjoint edges that cover all vertices of G (i.e., a perfect matching of G) is called a Kekulé pattern of G, and a set of pairwise disjoint cells of G such that the deletion of all vertices incident to these cells results in a graph that has a Kekulé pattern, or is empty, is called a Ciar pattern of G. Let k(G) and c(G) denote the number of Kekulé patterns and of Ciar patterns of G, respectively. It is shown that k(G) is not smaller than c(G) and that equality holds if G is an outerplane graph. This result generalizes a well-known proposition of the theory of benzenoid hydrocarbons; the proof uses a new idea.
rnThe quantum mechanical relevance of the concept of a spanning tree extant within a given molecular graph-specifically, one that may be considered to represent the carbon-atom connectivity of a particular (planar) conjugated system-was first explicitly pointed out by Professor Roy McWeeny in his now-classic 1958 memoire entitled "Ring Currents and Proton Magnetic Resonance in Aromatic Molecules." In a recent work, Gutman and one of the present authors proposed a scheme for calculating the number of spanning trees in the graph associated with catacondensed, benzenoid molecules which, by definition, contain rings of just the one size (six-membered); here, we present an algorithmic approach that enables the determination of the number of spanning trees in the molecular graph of any catacondensed system (which, in general, has rings of more than one size, and these may be of any size). An illustrative example is given, in which the algorithm devised is applied to a (hypothetical) pentacyclic catacondensed structure comprising a five-membered ring, a six-membered ring, a seven-membered ring, and two four-membered rings. 0 1996
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