Gain distance matrices for complex unit gain graphs

A complex unit gain graph ( T -gain graph) Φ = G , φ ) is a where the function assigns number to each orientation of an edge and its inverse assigned opposite orientation. balanced if product gains oriented cycle (if any) 1. We propose two notions distance matrices D < max min Φ, for any ordering ‘<’ vertex set. characterize graphs which are independent ordering. show holds standard vertices only same vertices, we call such as compatible graphs. whose cospectral with matrix underlying graph. Besides, introduce notion positively weighted establish equivalent condition balance Acharya's Stanić's spectral criteria deduced consequence. obtain some characterizations in terms matrices. Finally, bipartite every block compatible.