The diameters of commuting graphs of linear groups and matrix rings over the integers modulo m

2010/02/28 by Michael Giudici, Aedan Pope, Giudici, Michael +1
Mathematics · #05C25 (Primary) #05E15 (Secondary) #20H25 (Secondary) #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #math.CO #math.GR #msc:05C25 #msc:05E15 #msc:20H25

paper · pdf · doi:10.48550/arxiv.1003.0156

10 pages, v.3: introduction & bibliography corrected

openalex publication_date 2010/02/28 · arxiv created 2010/04/09 · arxiv updated 2010/04/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

The commuting graph of a group G, denoted by Gamma(G), is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. Let Z_m be the commutative ring of equivalence classes of integers modulo m. In this paper we investigate the connectivity and diameters of the commuting graphs of GL(n,Z_m) to contribute to the conjecture that there is a universal upper bound on diam(Gamma(G)) for any finite group G when Gamma(G) is connected. For any composite m, it is shown that Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) are connected and diam(Gamma(GL(n,Z_m))) = diam(Gamma(M(n,Z_m))) = 3. For m a prime, the instances of connectedness and absolute bounds on the diameters of Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) when they are connected are concluded from previous results.

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