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 on...

In 2005, Exoo posed the following question. Fix $\epsilon$ with $0\leq\epsilon&amp;lt;1$. Let $G_\epsilon$ be graph whose vertex set is Euclidean plane, where two vertices are adjacent iff distance between them lies in closed interval $[1-\epsilon,1+\epsilon]$. What chromatic number $\chi(G_\epsilon)$ of this graph? The case $\epsilon=0$ precisely classical ``chromatic plane'' problem. a 2018 p...