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 Trees: Theory and Applications

G. Clar 6n Rule versus Hückel 4n + 2 Rule 3464 H. Hydrocarbons versus Heteroatomic Systems 3465 IV. Hidden Treasures of Kekulé Valence Structures 3466 A. Conjugated Circuits 3467 B. Innate Degree of Freedom 3470 C. Clar Structures 3472 V. Graph Theoretical Approach to Chemical Structure 3473 A. Metric 3473 B. Chemical Graphs 3473 C. Isospectral Graphs 3473 D. Embedded Graphs 3475 E. Partial Ordering 3476 VI. On Enumeration of Benzenoid Hydrocarbons 3477 VII. Kekulé Valence Structures Count 3479 A. Non-branched Cata-condensed Benzenoids 3481 B. Branched Cata-condensed Benzenoids 3482 C. Benzenoid Lattices 3482 D. Peri-condensed Benzenoids 3483 E. Miscellaneous Benzenoids 3484 F. The Approach of Platt 3485 G. Computer Programs for Calculating K 3485 H. Transfer-Matrix Method 3486 I. Use of Recursion Relations 3486 J. Use of Signed Matrices 3487 VIII. Enumeration of Conjugated Circuits 3488 IX. Approximate Approaches versus Ambitious Computations 3490

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Aromaticity of polycyclic conjugated hydrocarbons.

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. Hexagonal systems (HS's) 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 HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's. A few open problems are mentioned. The chemical applications of the results presented are explained in detail.

https://www.fmf.uni-lj.si/~klavzar/preprints/Wiener-survey.pdf

Wiener Index of Hexagonal Systems

Centers of complex networks.

A word is just a finite string of letters. The word Wavoids the word U provided no substitution instance of Uis a subword of W. W is avoidable if on some finite alpha-bet there is an infinite collection of words each of whichavoids W. W is A th power-free if W avoids x , where x isa letter. We develope the theory of those endomorphismsof free semigroups which preserve A th power-freeness andemploy this theory to investigate &th power-free words.We go on to prove that every Mh power-free word on nletters is a subword of a maximal word of the same kind.Next we examine avoidable words in general and provethat all words of length at least 2

Douglas E. Jackson

Papers

Distance in graphs

On nonrepetitive sequences

Enumeration of certain binary matrices

A direct proof of the duality theorem of KÖNIG

Matrices permutable to ∗ matrices