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Abstract
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There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider commuting/nilpotent/solvable conjugacy class graph of where two distinct conjugacy classes and are adjacent if there exist some elements and such that is abelian/nilpotent/solvable. After a section of introductory results and examples, we discuss all the available results on connectedness, graph realization, genus, various spectra and energies of certain induced subgraphs of these graphs. Proofs of the results are not included. However, many open problems for further investigation are stated.
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