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On Integral Sum Labeling of Dense Graphs
並列摘要
A graph is said to be a sum graph if there exists a set S of positive integers as its vertex set with two vertices adjacent whenever their sum is in S. An integral sum graph is defined just as the sum graph, the difference being that the label set S is a subset of Z instead of set of positive integers. The sum number of a given graph G is defined as the smallest number of isolated vertices which when added to G results in a sum graph. The integral sum number of G is analogous. In this paper, we mainly prove that any connected graph G of order n with at least three vertices of degree (n-1) is not an integral sum graph. We characterise the integral sum graph G of order n having exactly two vertices of degree (n-1) each and hence give an alternative proof for the existence theorem of sum graphs.