The concept of the sum graph and integral sum graph were introduced by F. Harary. In this paper, we gain some upper and lower bounds on the sum number and the integral sum number of a graph and these bounds are sharp, and some new properties on the integral sum graph. Using these results, we could directly investigate and determine the exclusive integral sum numbers, the exclusive sum numbers, the sum numbers and the integral sum numbers of the graphs \(K_n\backslash E(2P_3)\), \(K_n\backslash E(P_3)\) and any graph \(H\) with minimum degree \(\delta(H) = n-2\) respectively as \(2\) is more than a given number. Then they will be the beginning of a new thought of research on the (exclusive) sum graph and the (exclusive) integral sum graph.
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