Ordering chemical unicyclic graphs by Wiener polarity index

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d _G (u, v) = 3}, where d _G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W _p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given.     

PI indices of pericondensed benzenoid graphs

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., , where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of a class of pericondensed benzenoid graphs consisting of three rows of hexagonal of various lengths.     

Generalized Wiener indices of zigzagging pentachains

The “pentachains” studied in this paper are graphs formed of concatenated 5-cycles. Explicit formulas are obtained for the Schultz and modified Schultz indices of these graphs, as well as for generalizations of these indices. In the process we give a more refined version of the procedure that earlier was reported for the ordinary Wiener index.     

Extremal Wiener and Kirchhoff indices of globular caterpillars

The Wiener and Kirchhoff indices of a graph G are two of the most important topological indices in mathematical chemistry. A graph G is called to be a globular caterpillar if G is obtained from a complete graph K _s with vertex set {v₁,v₂,…, v _s} by attaching n _i pendent edges to each vertex v _i of K _s for some positive integers s and n₁,n₂,…,n _s, denoted by . Let be the set of globular caterpillars with n vertices (). In this article, we characterize the globular caterpillars with the minimal and maximal Wiener and Kirchhoff indices among , respectively.     

Terminal Wiener index

Motivated by some recent research on the terminal (reduced) distance matrix, we consider the terminal Wiener index (TW) of trees, equal to the sum of distances between all pairs of pendent vertices. A simple formula for computing TW is obtained and the trees with minimum and maximum TW are characterized.     

Some upper bounds for the energy of graphs

Let G = (V,E) be a graph with n vertices and e edges. Denote V(G) = {v ₁,v ₂,...,v _n }. The 2-degree of v _i , denoted by t _i , is the sum of degrees of the vertices adjacent to . Let σ _i be the sum of the 2-degree of vertices adjacent to v _i . In this paper, we present two sharp upper bounds for the energy of G in terms of n, e, t _i , and σ _i , from which we can get some known results. Also we give a sharp bound for the energy of a forest, from which we can improve some known results for trees.     

The (n,n)-graphs with the first three extremal Wiener indices

Let G = (V, E) be a simple connected graph with vertex set V and edge set E. The Wiener index W(G) of G is the sum of distances between all pairs of vertices in G, i.e., , where d _G (u, v) is the distance between vertices u and v in G. In this paper, we first give a new formula for calculating the Wiener index of an (n,n)-graph according its structure, and then characterize the (n,n)-graphs with the first three smallest and largest Wiener indices by this formula.     

Computing weighted Szeged and PI indices from quotient graphs

The weighted (edge-)Szeged index and the weighted (vertex-)PI index are modifications of the (edge-)Szeged index and the (vertex-)PI index, respectively, because they take into account also the vertex degrees. As the main result of this article, we prove that if G is a connected graph, then all these indices can be computed in terms of the corresponding indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than the Θ^*-partition. If G is a benzenoid system or a phenylene, then it is possible to choose a partition of the edge set in such a way that the quotient graphs are trees. As a consequence, it is shown that for a benzenoid system, the mentioned indices can be computed in sublinear time with respect to the number of vertices. Moreover, closed formulas for linear phenylenes are also deduced.     

On Acyclic Molecular Graphs with Prescribed Numbers of Edges that Connect Vertices with given Degrees

We find a necessary and sufficient conditions on a sequence

Conjugated chemical trees with minimal energy and prescribed diameter

The energy of a molecular graph G is defined as the sum of the absolute values of the eigenvalues of A(G), where A(G) is the adjacency matrix of this graph. This article characterizes conjugated chemical trees with prescribed diameter and minimal energies and presents explicit expressions of their Hosoya indices. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix

We report some properties of the maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix of a connected graph, in particular, various lower and upper bounds, and the Nordhaus–Gaddum‐type results for them. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Studies of some monocyclic and bicyclic arylacetonitriles

The high-resolution (1)H, (13)C, (1)H-(1)H COSY and (1)H-(13)C COSY NMR spectra have been recorded in CDCl(3) for arylacetonitriles 1-12 and analyzed. The arylacetonitriles 3-7 exist in two isomeric forms E (methyl group is anti to cyano group) and Z (the methyl group is syn to cyano group) in solution. Normal chair conformation with equatorial orientations of phenyl rings at C-2 and C-6 for monocyclic nitriles 1 and 2, epimeric chair structure EC (axial configuration of methyl group at C-3) for both the E and Z isomers of arylacetonitrile derivatives (3-7) and a distorted boat form, B(3), for the N-acylacetonitrile derivatives (8-10) have been proposed based on NMR data. The bicyclic nitriles 11 and 12 exist in twin chair conformations in solution. DFT calculations and chemical shifts also support these conformations. Geometry optimizations for 1-12 were carried out according to density functional theory using B3LYP/6-31G(d,p) basis set and for 1 and 8 the theoretical geometrical parameters have been compared with those of single crystal measurements.     

Wiener polynomials of some chemically interesting graphs

The chemist Harold Wiener found ??(G), the sum of distances between all pairs of vertices in a connected graph G, to be useful as a predictor of certain physical and chemical properties. The q‐analogue of ??, called the Wiener polynomial ??(G; q), is also useful, but it has few existing useful formulas. We will evaluate ??(G; q) for certain graphs G of chemical interest. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Connections between Wiener index and matchings

Let T be an acyclic molecule with n vertices, and let S(T) be the acyclic molecule obtained from T by replacing each edge of T by a path of length two. In this work, we show that the Wiener index of T can be explained as the number of matchings with n−2 edges in S(T). Furthermore, some related results are also obtained MSC: 05C12 Weigen Yan: This work is supported by FMSTF (2004J024) and NSFF(E0540007) Yeong-Nan Yeh: Partially supported by NSC94-2115-M001-017