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 ̸= 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.
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