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This e-book addresses a brand new interdisciplinary quarter rising at the border among a number of components of arithmetic, physics, chemistry, nanotechnology, and desktop technology. the point of interest this is on difficulties and strategies concerning graphs, quantum graphs, and fractals that parallel these from differential equations, differential geometry, or geometric research. additionally incorporated are such assorted subject matters as quantity thought, geometric workforce concept, waveguide idea, quantum chaos, quantum cord structures, carbon nano-structures, metal-insulator transition, desktop imaginative and prescient, and communique networks. This quantity encompasses a targeted choice of professional experiences at the major instructions in research on graphs (e.g., on discrete geometric research, zeta-functions on graphs, lately rising connections among the geometric team thought and fractals, quantum graphs, quantum chaos on graphs, modeling waveguide platforms and modeling quantum graph structures with waveguides, regulate thought on graphs), in addition to learn articles.
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Additional resources for Analysis on Graphs and Its Applications
Oellermann, Menger-type results for three or more vertices, Congr. Numer. 113 (1996), 179–204. 38. A. Kaneko and K. Ota, On minimally (n, λ)-connected graphs, J. Combin. Theory (B) 80 (2000), 156–171. 39. J. Keijsper and A. Schrijver, On packing connectors, J. Combin. Theory (B) 73 (1998), 184–188. 40. D. König, Graphok és matrixok, Mat. Fiz. Lapok 38 (1931), 116–119. 41. D. König, Über trennende Knotenpunkte in Graphen, Acta Litt. Sci. Szeged 6 (1933), 155–179. 38 Ortrud R. Oellermann 42. D. König, Theorie der Endlichen und Unendlichen Graphen, Chelsea, 1950.
Section 6 is devoted to Menger-type results for paths of bounded length between a pair of vertices. In particular, we discuss relationships between the maximum number of internally disjoint paths of a constrained length connecting a pair of vertices and the minimum number of vertices whose deletion destroys all such paths between this pair of vertices. We conclude the chapter with two sections that survey Menger-type results for sets of more than two vertices. Throughout most of the chapter, we focus on vertex results, but usually, as with Menger’s theorem itself, there is a corresponding edge result; when appropriate, we also discuss these.
W v Fig. 6. 1 If d is the distance between vertices v and w in graph G, then κd (v, w) = μd (v, w). We now turn to upper bounds for κd (v, w) in terms of μd (v, w). It is convenient to look at these in the form of the ratio ρd (v, w) = κd (v, w) . μd (v, w) The following results were established in . 2 If v and w are non-adjacent vertices in a graph G, then (a) ρ2 (v, w) = ρ3 (v, w) = ρ4 (v, w) = 1; (b) ρ5 (v, w) = 2; (c) for d ≥ 6, ρd (v, w) ≤ 12 d . It is not known, for general d, how close ρd (v, w) can get to 12 d.