Balance theory: An extension to conjugate skew gain graphs

We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the g-Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs.