2,584 research outputs found

    Monotone Maps, Sphericity and Bounded Second Eigenvalue

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    We consider {\em monotone} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on nn points can be embedded into l2nl_2^n, while, (in a sense to be made precise later), for almost every nn-point metric space, every monotone map must be into a space of dimension Ω(n)\Omega(n). It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on {\em sphericity} of graphs, which suggest one example of such a metric space - that defined by a complete bipartitegraph. We prove that an Ύn\delta n-regular graph of order nn, with bounded diameter has sphericity Ω(n/(λ2+1))\Omega(n/(\lambda_2+1)), where λ2\lambda_2 is the second largest eigenvalue of the adjacency matrix of the graph, and 0 < \delta \leq \half is constant. We also show that while random graphs have linear sphericity, there are {\em quasi-random} graphs of logarithmic sphericity. For the above bound to be linear, λ2\lambda_2 must be constant. We show that if the second eigenvalue of an n/2n/2-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only o(n2)o(n^2) entries. Furthermore, for any 0 < \delta < \half, and λ2\lambda_2, there are only finitely many Ύn\delta n-regular graphs with second eigenvalue at most λ2\lambda_2

    Regular embeddings of complete bipartite maps: classification and enumeration

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    The regular embeddings of complete bipartite graphs Kn, n in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where n is a prime power, obtained in collaboration with Du, Kwak, Nedela and koviera, together with results of ItĂŽ, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups. <br/

    On the orientable regular embeddings of complete multipartite graphs

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    AbstractLet Km[n] be the complete multipartite graph with m parts, while each part contains n vertices. The regular embeddings of complete graphs Km[1] have been determined by Biggs (1971) [1], James and Jones (1985) [12] and Wilson (1989) [23]. During the past twenty years, several papers such as Du et al. (2007, 2010) [6,7], Jones et al. (2007, 2008) [14,15], Kwak and Kwon (2005, 2008) [16,17] and Nedela et al. (2002) [20] contributed to the regular embeddings of complete bipartite graphs K2[n] and the final classification was given by Jones [13] in 2010. Since then, the classification for general cases m≄3 and n≄2 has become an attractive topic in this area. In this paper, we deal with the orientable regular embeddings of Km[n] for m≄3. We in fact give a reduction theorem for the general classification, namely, we show that if Km[n] has an orientable regular embedding M, then either m=p and n=pe for some prime p≄5 or m=3 and the normal subgroup Aut0+(M) of Aut+(M) preserving each part setwise is a direct product of a 3-subgroup Q and an abelian 3â€Č-subgroup, where Q may be trivial. Moreover, we classify all the embeddings when m=3 and Aut0+(M) is abelian. We hope that our reduction theorem might be the first necessary approach leading to the general classification

    Complete bipartite graphs whose topological symmetry groups are polyhedral

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    We determine for which nn, the complete bipartite graph Kn,nK_{n,n} has an embedding in S3S^3 whose topological symmetry group is isomorphic to one of the polyhedral groups: A4A_4, A5A_5, or S4S_4.Comment: 25 pages, 6 figures, latest version has minor edits in preparation for submissio
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