The commuting graph of a group G, denote d by Γ( G), is a simple undirected graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = y x. The commuting graph of a subset of a group is defined similarly. In th is paper we investigate the properties of the commuting graph of the Symmetric and Alternating and subsets of transpositions and involutions in the Symmetric groups.

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