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A graph theoretic approach for identification of such customer in a communication network is proposed. The presence of interior node and end node in a fuzzy graph and their properties are discussed based on sum distance metric. The effect of interior node and end node on blocks is analyzed. The condition for existence of interior node in complete fuzzy graph is discussed. Interior node acts as a sensory node which loses its property if there is a decrease in number of outgoing calls or duration of calls decreases. 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