Abstract
Questions about a graph’s connected components are answered by studying appropriate powers of a special “adjacency matrix” constructed with entries in a commutative algebra whose generators are idempotent. The approach is then applied to the Erdös–Rényi model of sequences of random graphs. Developed herein is a method of encoding the relevant information from graph processes into a “second quantization” operator and using tools of quantum probability and infinite-dimensional analysis to derive formulas that reveal the exact values of quantities that otherwise can only be approximated. In particular, the expected size of a maximal connected component, the probability of existence of a component of particular size, and the expected number of spanning trees in a random graph are obtained.
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Schott, R., Staples, G.S. Connected components and evolution of random graphs: an algebraic approach. J Algebr Comb 35, 141–156 (2012). https://doi.org/10.1007/s10801-011-0297-1
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DOI: https://doi.org/10.1007/s10801-011-0297-1