Abstract
Evolving random graph R ∞ as introduced and studied by Erdös and Rényi, represent the limit of random f-graphs R f when f→∞. The latter have been studied mainly by chemists. Such systems show an abrupt transition with the appearance of a giant component (1-connected subgraph) which models transitions in physical systems. It is known that further abrupt transitions in R f (and R ∞) occur with the appearance of giant k-connected subgraphs and these transitions also appear to have their counterparts in physical systems. Cycle length distributions in R f and R ∞ (following the k=1 transition) appear to be inconsistent with the use of these random graphs as physical models and have led to the use of random lattice-graphs R L(f). Results from percolation theory in physics relate to an abrupt transition for 1-connected subgraphs and lead to some interesting conclusions about the use of random lattice-graphs when these systems are compared to the transition in R f. Further progress in applying random graph theory to model physical systems requires that similar results on abrupt transitions for k-connected subgraphs and on cycle distributions in R L(f) be obtained. We report here on progress and problems of this type in the setting of the applicability of random graphs to model highly interesting physical, chemical and biological systems.
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Bloom, G.S., Mandziuk, M.T., Kennedy, J.W., Quintas, L.V. (1983). Random graphs and the physical world. In: Borowiecki, M., Kennedy, J.W., Sysło, M.M. (eds) Graph Theory. Lecture Notes in Mathematics, vol 1018. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071618
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DOI: https://doi.org/10.1007/BFb0071618
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