b-Coloring of G is a coloring which is proper such that in each color class there exists a vertex which is called as representative vertex that has at least one neighbor in each of the remaining color classes. The highest positive integer k such that the k-colors can be used to color the vertices of G along with b-coloring is the b-chromatic number of G, denoted by b(G). For a given graph G with n vertices, G ∗ is constructed (Jeeva et al., Indian J Math 59(2):255–261, 2017). In the research paper, we find out the b-chromatic number of Mycielskian, splitting, shadow, middle, and total graph of G ∗ . © Springer Nature Switzerland AG 2019.