Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 9 2 2024 06 01 Balance theory: An extension to conjugate skew gain graphs 253 262 14466 10.22049/cco.2022.27933.1397 EN Shahul Hameed Koombail Department of Mathematics, K M M Government Women’s College, Kannur - 670004, Kerala, India Ramakrishnan K O Department of Mathematics, K M M Government Women’s College, Kannur - 670004, Kerala, India Journal Article 2022 07 23 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. https://comb-opt.azaruniv.ac.ir/article_14466_8929c6504b7a82b6b7a07c3e5da470f8.pdf