on some properties of edge quasi-distance-balanced graphs

For an edge e = uv in a graph g, mg u (e) is introduced as the set all edges of g that are at shorter distance to u than to v. we say that g is an edge quasi-distance-balanced graph whenever for every arbitrary edge e = uv, there exists a constant λ > 1 such that mg u (e) = λ ±1mg v (e). we investigate that edge quasi-distance-balanced garphs are complete bipartite graphs km,n with m 6= n. the aim of this paper is to investigate the notion of cycles in edge quasi-distance-balanced graphs, and expand some techniques generalizing new outcome that every edge quasi-distance-balanced graph is complete bipartite graph. as well as, it is demontrated that connected quasi-distance-balanced graph admitting a bridge is not edge quasi-distance-balanced graph.