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Undirected Graphs Realizable as Graphs of Modular Lattices

Published online by Cambridge University Press:  20 November 2018

Laurence R. Alvarez
Laurence R. Alvarez*
Affiliation:
University of the South, Sewanee, Tennessee
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If (L, ≥) is a lattice or partial order we may think of its Hesse diagram as a directed graph, G, containing the single edge E(c, d) if and only if c covers d in (L, ≥). This graph we shall call the graph of (L, ≥). Strictly speaking it is the basis graph of (L, ≥) with the loops at each vertex removed; see (3, p. 170).

We shall say that an undirected graph Gu can be realized as the graph of a (modular) (distributive) lattice if and only if there is some (modular) (distributive) lattice whose graph has Gu as its associated undirected graph.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 923 - 932
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Birkhoff, G., Lattice theory, 1st éd. (Providence, R.I., 1940).Google Scholar
2. Birkhoff, G., Lattice theory, rev. ed. (Providence, R.I., 1961).Google Scholar
3. Ore, O., Theory of graphs (Providence, R.I., 1962).Google Scholar