Andrey A. Dobrynin

Wiener Index of Trees: Theory and Applications

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.

Wiener Index of Hexagonal Systems

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. Hexagonal systems (HSs) are a special type of plane graphs in which all faces are bounded by hexagons. These provide a graph representation of benzenoid hydrocarbons and thus find applications in chemistry. The paper outlines the results known for W of the HS: method for computation of W, expressions relating W with the structure of the respective HS, results on HSs extremal w.r.t. W, and on integers that cannot be the W-values of HSs. A few open problems are mentioned. The chemical applications of the results presented are explained in detail.

TREES WITH EXTREMAL HYPER-WIENER INDEX : MATHEMATICAL BASIS AND CHEMICAL APPLICATIONS

Trees with minimal and maximal hyper-Wiener indices (WW) are determined:  Among n-vertex trees, minimum and maximum WW is achieved for the star-graph (Sn) and the path-graph (Pn), respectively. Since WW(Sn) is a quadratic polynomial in n,, whereas WW(Pn) is a quartic polynomial in n, the hyper-Wiener indices of all n-vertex trees assume values from a relatively narrow interval. Consequently, the hyper-Wiener index must have a very low isomer-discriminating power. This conclusion is corroborated by finding large families of trees, all members of which have equal WW-values.

Wiener index for graphs and their line graphs with arbitrary large cyclomatic numbers

Abstract The Wiener number, W ( G ) , is the sum of the distances of all pairs of vertices in a graph G . Infinite families of graphs with increasing cyclomatic number and the property W ( G ) = W ( L ( G ) ) are presented, where L ( G ) denotes the line graph of G . This gives a positive (partial) answer to an open question posed in an earlier paper by Gutman, Jovasevic, and Dobrynin.

A Wiener-type graph invariant for some bipartite graphs

Abstract In this paper, a Wiener-type graph invariant W ∗ is considered, defined as the sum of the product nu(e)nv(e) over all edges e = (u, v) of a connected graph G, where nu(e) is the number of vertices of G, lying closer to u than to v. A class C(h, k) of bipartite graphs with cyclomatic number h is designed, such that for G1, G2 ∈ C(h, k), W ∗ (G 1 ) ≡ W ∗ (G 2 ) ( mod 2k 2 ) . This fully parallels a previously known result for the Wiener number.

Some results on the Wiener index of iterated line graphs

Abstract The Wiener index W ( G ) of a graph G is the sum of distances between all unordered pairs of vertices. This notion was motivated by various mathematical properties and chemical applications. For a tree T, it is known that W ( T ) and W ( L ( T ) ) are always distinct. It is shown that there is an infinite family of trees with W ( T ) = W ( L 2 ( T ) ) .

A simple formula for the calculation of of the wiener index of hexagonal chains

Abstract The Wiener index (W) of hexagonal chains (i.e. the molecular graphs of unbranched catacondensed benzenoid hydrocarbons) is examined. The index W is a topological index defined as the sum of distances between all pairs of vertices in a molecular graph. A simple calculation formula for W is put forward. As a corollary, necessary and sufficient conditions for the coincidence of W-values of certain hexagonal chains are established.

THE AVERAGE WIENER INDEX OF TREES AND CHEMICAL TREES

Meir and Moon (J. Combin. Theory 1970, 8, 99−103) reported a combinatorial formula for the average value of the distance between a pair of vertices in the class of all labeled trees with a fixed number (= n) of vertices. From this result an expression for the average Wiener index 〈Wn〉lab of labeled n-vertex trees follows immediately. We show that both the average Wiener index 〈Wn〉 of nonlabeled n-vertex trees and the average Wiener index 〈Wn〉ch of nonlabeled n-vertex chemical trees having n ≤ 20 vertices are proportional to 〈Wn〉1ab, with proportionality constants around 0.927 and 0.990, respectively. Analogous results are obtained for the Hosoya polynomial.

The average Wiener index of hexagonal chains

Abstract The average value of the Wiener index of hexagonal chains (= unbranched catacondensed benzenoid systems) with a fixed number of hexagons, is calculated. This average value differs somewhat from what earlier was reported (Gutman et al., 1990. Chem. Phys. Lett. 173, 403). The origin of this difference is clarified. A remarkable collective property is established for the chains possessing no linearly annelated hexagons: the sum of their Wiener indices is divisible by the number of chains.

Regular graphs having the same path layer matrix

The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all paths in G. Elements (i, j) in this matrix is the number of simple paths in G having initial vertex v i and length j. For every r≥3, pairs of nonisomorphic r-regular graphs having the same path layer matrix are presented