Author
Roger C. Entringer
Bio: Roger C. Entringer is an academic researcher from University of New Mexico. The author has contributed to research in topics: Pancyclic graph & Indifference graph. The author has an hindex of 10, co-authored 19 publications receiving 1648 citations.
Papers
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TL;DR: The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph as discussed by the authors, defined as the distance between all vertices in a graph.
Abstract: 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.
1,015 citations
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360 citations
TL;DR: There exists an infinite binary sequence having no identical adjacent blocks of length 3 or greater and every infinitebinary sequence has arbitrarily long adjacent blocks that are permutations of each other.
98 citations
TL;DR: In this article, it was shown that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most (n 2 + 1 + 5 ) + 5.
Abstract: The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ( n2)-1 Σ{x,y}‚V(G) dG(x, y), where V(G) denotes the vertex set of G and dG(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most $${n\over \delta + 1} + 5$$. We give improved bounds for K3-free graphs, C4-free graphs, and for graphs of given girth. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 113, 2000
58 citations
TL;DR: Every induced subgraph of the n-cube, Qn, with more than ⌈2n+13⌉ vertices is shown to contain a 4-cycle; this bound is sharp.
54 citations
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TL;DR: Gossiping and broadcasting are two problems of information dissemination described for a group of individuals connected by a communication network as discussed by the authors, and the results that have been obtained on these and related problems.
Abstract: Gossiping and broadcasting are two problems of information dissemination described for a group of individuals connected by a communication network. In gossiping every person in the network knows a unique item of information and needs to communicate it to everyone else. In broadcasting one individual has an item of information which needs to be communicated to everyone else. We review the results that have been obtained on these and related problems.
1,191 citations
TL;DR: The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph as discussed by the authors, defined as the distance between all vertices in a graph.
Abstract: 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.
1,015 citations
TL;DR: Theoretical Approach to Chemical Structure, Approximate Approaches versus Ambitious Computations, and Use of Signed Matrices.
Abstract: 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
664 citations
TL;DR: A survey of existing methods of communication in usual networks, particularly the complete network, the ring, the torus, the grid, the hypercube, the cube connected cycles, the undirected de Bruijn graph, the stargraph, the shuffle-exchange graph, and the butterfly graph.
398 citations
TL;DR: In this paper, the authors present the results known for W of the HS: method for computing 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.
Abstract: 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.
371 citations