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The energy of a graph of a group G is the sum of all absolute values of the eigenvalues of the adjacency matrix. An adjacency matrix is a square matrix where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. A conjugacy class graph is a graph whose vertex set is the non-central conjugacy classes of the group. Two vertices are connected if their orders are not coprime. Meanwhile, a group G is said to be metabelian if there exists a normal subgroup H in G such that both H and the factor group G/H are abelian. In this research, the energy of the conjugacy class graphs for all nonabelian metabelian groups of order 24 are determined. The computations are assisted by Groups, Algorithm and Programming (GAP) and Maple2016 software. The results show that the energy of graphs of the groups in the study must be an even integer in the case that the energy is rational. [ABSTRACT FROM AUTHOR]

In group theory, conjugacy class is a method of partitioning the elements of a group such that the elements a and b are conjugate in a group G if xax-1 = b for some x in G. Meanwhile, a point group is a set of symmetry operations that keeps at least one point in a molecule fixed. In chemistry, symmetry of molecules is important since chemists classify molecules based on their symmetry. In this research, the conjugacy classes of point groups of order at most eight are computed. The conjugacy classes of these groups are then applied to graph theory to obtain the conjugacy class graph, which is a graph whose vertices V = (v1,...,vn) are the non-central conjugacy classes of a group and the two vertices are connected if their cardinalities are not coprime, that is the greatest common divisor of the cardinalities of the corresponding vertices is not equal to one. [ABSTRACT FROM AUTHOR]