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In this paper, the use of collapsed adjacency diagrams and collapsed adjacency matrices on the commuting graph 𝒞 (G, X) is discussed. Here, we demonstrate the concepts on the symmetric group of degree n = 3r, Sym(n) = Sym(3r) where r, the number of 3-cycles in the group element t, is either 3 or 4. [ABSTRACT FROM AUTHOR]

In this paper, we aim to compute the energy of commuting graph C(G,X)of symplectic groups for conjugacy classes of involution and elements of order three by using the eigenvalues of the corresponding collapsed adjacency matrices. In the last section we also highlight an interesting observation from the energy of C(G,X) in symplectic groups. [ABSTRACT FROM AUTHOR]

The commuting graph C(G,X), where G is a finite group and X is a subset of G, is the graph whose vertex set is X and two distinct elements of X being joined by an edge whenever they commute in the group G. Here the Δ1 (t)-orbit representatives and the number of elements in the first disc Δ1 (t) of Δ1 (t), is studied when G is a symmetric group of degree n, Sym(n) and X is a conjugacy class of elements of order three. [ABSTRACT FROM AUTHOR]