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A q-analogous of the characterization of hypercubes as graphs

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Summary

We give an arithmetical characterization of graphs which are realizable as graphs of the lattice ℒ of the subspaces of a graphic (or projective) space of finite dimension and of finite order q ≥ 1. In other words ℒ is any complemented modular lattice of finite rank and of finite order q. When q ≥ 2 and the rank is at least four, ℒ is the lattice of the subspaces of a finite dimensional vector space over the field GF(q). Two independent axioms are required involving the number of geodesies between any two vertices. This number must be a simple function of the distance.

Our characterization may be consider as a q-analogous of the characterization of hypercubes, given by S.Foldes [6], obtained here for q=1 in a slightly different form.

After the formulation of the two axioms (n.1) and some recalls on graphic spaces and on projective spaces (n.2), we study the graph of their subspaces establishing the necessity of the axioms (n.3). The sufficiency and the independence of the axioms are obtained in n.4.

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Authors and Affiliations

  1. Dipartimento di Matematica, “G. Castelnuovo” Città Universitaria, 00185, Roma

    P. V. Ceccherini

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  1. P. V. Ceccherini
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Dedicated to Professor Adriano Barlotti on the occasion of his 60-th birthday.

Work partially supported by GNSAGA (Section 4) of the Italian CNR. Communicated at University of Bordeaux (June 1982) and at Oberwolfach (Gründlagen der Geometrie, October 1982).

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Ceccherini, P.V. A q-analogous of the characterization of hypercubes as graphs. J Geom 22, 57–74 (1984). https://doi.org/10.1007/BF01230125

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