Given a finite group G, the coprime graph of G, denoted by Γ(G), is defined as an undirected graph with the vertex set G, and for distinct x, y ∈ G, x is adjacent to y if and only if (o(x), o(y)) = 1, where o(x) and o(y) are the orders of x and y, respectively. This paper classifies the finite groups with split, threshold and chordal coprime graphs, as well as gives a characterization of the finite groups whose coprime graph is a cograph. As some applications, the paper classifies the finite groups G such that Γ(G) is a cograph if G is a nilpotent group, a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.
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