Author
Jack E. Graver
Other affiliations: Dartmouth College, Purdue University, Emory University
Bio: Jack E. Graver is an academic researcher from Syracuse University. The author has contributed to research in topics: Fullerene & Planar graph. The author has an hindex of 13, co-authored 49 publications receiving 983 citations. Previous affiliations of Jack E. Graver include Dartmouth College & Purdue University.
Papers
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01 Sep 1971
TL;DR: In this paper, the generalized Petersen graph G(n, k) was defined for integers n and k with 2 ≤ 2k < n, and G(5, 2) is the well known Petersen graph.
Abstract: 1. Introduction. For integers n and k with 2 ≤ 2k < n, the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-setand edge-set E(G(n, k)) to consist of all edges of the formwhere i is an integer. All subscripts in this paper are to be read modulo n, where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2n, and G(5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)
190 citations
TL;DR: Borders for μ(Γ) are computed in terms of the number of vertices in Γ and the diameter of Γ to prove a formula for computing μ( Γ) whenΓ is a tree which is particularly useful when Γ has a high degree of symmetry.
156 citations
TL;DR: This paper gives a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms, and proves that any “improvement process” which searches through a test set at each stage converges to an optimal point in a finite number of steps.
Abstract: In this paper we consider the question: how does the flow algorithm and the simplex algorithm work? The usual answer has two parts: first a description of the "improvement process", and second a proof that if no further improvement can be made by this process, an optimal vector has been found. This second part is usually based on duality a technique not available in the case of an arbitrary integer programming problem. We wish to give a general description of "improvement processes" which will include both the simplex and flow algorithms, which will be applicable to arbitrary integer programming problems, and which will in themselves assure convergence to a solution. Geometrically both the simplex algorithm and the flow algorithm may be described as follows. At the i th stage, we have a vertex (or feasible flow) to which is associated a finite set of vectors, namely the set of edges leaving that vertex (or the set of unsaturated paths). The algorithm proceeds by searching among this special set for a vector along which the gain function is increasing. If such a vector is found, the algorithm continues by moving along this vector as far as is possible while still remaining feasible. The search is then repeated at this new feasible point. We give a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms. We will then prove that any "improvement process" which searches through a test set at each stage converges to an optimal point in a finite nmnber of steps. We also construct specific test sets which are the natural extensions of the test sets employed by the flow algorithm to arbitrary linear and integer linear programming problems.
154 citations
TL;DR: The purpose is to give a unified development of enumerative techniques which give sharp upper bounds on Ramsey's theorem numbers and to give constructive methods for partitions to determine lower bounds on these numbers.
110 citations
TL;DR: The existence problem for t-designs with prescribed parameters is solved by allowing positive and negative integral multiplicities for the blocks.
104 citations
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17 Dec 1994
TL;DR: In this article, the Conjectures of Hadwiger and Hajos are used to define graph types, such as planar graph, graph on higher surfaces, and critical graph.
Abstract: Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.
1,380 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
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TL;DR: This work presents data which, to the best of its knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge.
Abstract: We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values.
581 citations
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TL;DR: The theory of graphs has broad and important applications, because so many things can be modeled by graphs, and various puzzles and games are solved easily if a little graph theory is applied.
Abstract: A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.
541 citations