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(v, k, λ)-Graphs and polarities of (v, k, λ)-designs

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Abstract

In 2.1 it is established that there is a one-to-one correspondence between (v, k, λ)-graphs and polarities, with no absolute points, of (v, k, λ)-designs. This is used to show that the parameters of a (v, k, λ)-graph are of the form ((s/a)((s + a)2−1), s(s+a), sa) where s and a are positive integers with a dividing s(s2−1) (Theorem 3.4) but strictly less than s(s2−1) (Proposition 4.3). Some consequences of this parametrization are discussed and in particular, it is shown that for fixed 2 there are only finitely many non-isomorphic (v, k, λ)-graphs. In 4. it is shown that (v, k, λ)-graphs can also be constructed using polarities, with all points absolute, of certain designs. In 5. isomorphisms and automorphisms of graphs and designs are discussed. Many examples of (v, k, λ)-graphs, including some apparently new ones, are given.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, 48823, East Lansing, Michigan, USA

    Arunas Rudvalis

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  1. Arunas Rudvalis
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Dedicated to Peter Dembowski, † 28 January 1971

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Rudvalis, A. (v, k, λ)-Graphs and polarities of (v, k, λ)-designs. Math Z 120, 224–230 (1971). https://doi.org/10.1007/BF01117497

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  • DOI: https://doi.org/10.1007/BF01117497

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