Abstract
One finds a relationship between the theory of unimodular lattices and the theory of strongly regular graphs. For a unimodular, even lattice of dimension 32 having a system of roots of type A1 one constructs a strongly regular graph with parameters n=8184, a=7595, c=7042, d=7130. The graphs that arise from certain “Steiner sixtuple systems” have the same parameters. One also constructs strongly regular graphs for extremal lattices of dimension 48.
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Literature cited
B. B. Venkov, “On even unimodular Euclidean lattices of dimension 32,” Zap. Nauchn. Sem. LOMI, Vol.116, 44–55 (1982).
B. B. Venkov, “On even unimodular extremal lattices,” Trudy Mat. Inst. Akad. Nauk SSSR,165, 33–47, (1983).
P. J. Cameron and J. H. Van Lint, Graph Theory, Coding Theory and Block Designs, Cambridge Univ. Press, Cambridge (1975).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 30–38, 1983.
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Venkov, B.B. Unimodular lattices and strongly regular graphs. J Math Sci 29, 1121–1127 (1985). https://doi.org/10.1007/BF02106870
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DOI: https://doi.org/10.1007/BF02106870