Gain distance matrices for complex unit gain graphs

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