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| 2. |
- Huber, KT, et al.
(författare)
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The tight span of an antipodal metric space: Part II - Geometrical properties
- 2004
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Ingår i: Discrete & Computational Geometry. - 0179-5376. ; 31:4, s. 567-586
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Tidskriftsartikel (refereegranskat)abstract
- Suppose that X is a finite set and let R-x denote the set of functions that map X to R. Given a metric d on X, the tight span of (X, d) is the polyhedral complex T (X, d) that consists of the bounded faces of the polyhedronP(X, d) := {f is an element of R-x : f(x) + f (y) greater than or equal to d(x, y)}.In a previous paper we commenced a study of properties of T(X, d) when d is antipodal, that is, there exists an involution sigma : X --> X: x --> (x) over bar so that d(x, y) + d(y,(x) over bar) = d(x, (x) over bar) holds for all x, y c X. Here we continue our study, considering geometrical properties of the tight span of an antipodal metric space that arise from a metric with which the tight span comes naturally equipped. In particular, we introduce the concept of cell-decomposability for a metric and prove that the tight span of such a metric is the union of cells, each of which is isometric and polytope isomorphic to the tight span of some antipodal metric. In addition, we classify the antipodal cell-decomposable metrics and give a description of the polytopal structure of the tight span of such a metric.
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| 4. |
- Koolen, JH, et al.
(författare)
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Hyperbolic bridged graphs
- 2002
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Ingår i: European Journal of Combinatorics. - : Elsevier BV. - 0195-6698 .- 1095-9971. ; 23:6, s. 683-699
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Tidskriftsartikel (refereegranskat)abstract
- Given a connected graph G, we take, as usual, the distance xy between any two vertices x, y of G to be the length of some geodesic between x and y. The graph G is said to be delta-hyperbolic, for some 3 : 0, if for all vertices x, y, u, v in G the inequality xy + uv :5 max{xu + yv, xv + yu} + delta holds, and G is bridged if it contains no finite isometric cycles of length four or more. In this paper, we will show that a finite connected bridged graph is 1-hyperbolic if and only if it does not contain any of a list of six graphs as an isometric subgraph.
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| 5. |
- Koolen, JH, et al.
(författare)
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On a conjecture of Bannai and Ito: There are finitely many distance-regular graphs with degree 5, 6 or 7
- 2002
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Ingår i: European Journal of Combinatorics. ; 23:8, s. 987-1006
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Tidskriftsartikel (refereegranskat)abstract
- Bannai and Ito conjectured in a 1987 paper that there are finitely many distance-regular graphs with fixed degree that is greater than two. In a series of papers they showed that their conjecture held for distance-regular graphs with degrees 3 or 4. In this paper we prove that the Bannai-Ito conjecture holds for degrees 5-7.
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| 6. |
- Koolen, JH, et al.
(författare)
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The structure of spherical graphs
- 2004
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Ingår i: European Journal of Combinatorics. - 0195-6698. ; 25:2, s. 299-310
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Tidskriftsartikel (refereegranskat)abstract
- A spherical graph is a graph in which every interval is antipodal. Spherical graphs are an interesting generalization of hypercubes (a graph G is a hypercube if and only if G is spherical and bipartite). Besides hypercubes, there are many interesting examples of spherical graphs that appear in design theory, coding theory and geometry e.g., the Johnson graphs, the Gewirtz graph, the coset graph of the binary Golay code, the Gosset graph, and the Schlafli graph, to name a few. In this paper we study the structure of spherical graphs. In particular, we classify a subclass of these graphs consisting of what we call the strongly spherical graphs. This allows us to prove that if G is a triangle-free spherical graph then any interval in G must induce a hypercube, thus providing a proof for a conjecture due to Berrachedi, Havel and Mulder.
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