ON SOME PROPERTIES of EDGE QUASI-DISTANCE-BALANCED GRAPHS

Abstract

For an edge e = uv in a graph G, MGu (e) is introduced as the set all edgesof G that are at shorter distance to u than to v. We say that G is an edgequasi-distance-balanced graph whenever for every arbitrary edge e = uv,there exists a constant λ > 1 such that mGu(e) = λ±1mGv(e). We investigatethat edge quasi-distance-balanced garphs are complete bipartite graphsKm,n with m ̸= n. The aim of this paper is to investigate the notion of cyclesin edge quasi-distance-balanced graphs, and expand some techniquesgeneralizing new outcome that every edge quasi-distance-balanced graphis complete bipartite graph. As well as, it is demontrated that connectedquasi-distance-balanced graph admitting a bridge is not edge quasi-distance-balanced graph.