The integral sum number of complete bipartite graphs \(K_{r,s}\) (Q5946754)

A graph \(G=(V,E)\) is said to be an integral sum graph (sum graph) if its vertices can be given a labeling with distinct integers (positive integers), so that \(uv\in E\) if and only if \(u+v\in V\). The integral sum number (sum number) of a given graph \(G\) is the smallest number of isolated vertices \(S\) such that \(G\cup S\) is an integral sum graph (sum graph). The authors study and solve the problem of determining the integral sum number and the sum number of the complete bipartite graphs \(K_{r,s}\). NEWLINENEWLINENEWLINEThe results of this paper are strongly related to the results by \textit{W. Yan} and \textit{B. Liu} [Discrete Math. 240, No. 1-3, 219-229 (2001; Zbl 0983.05073)].