On \(\gamma\)-Labelings of Complete Bipartite Graphs

Abstract

Let \(G\) be a connected graph with \(p\) vertices and \(q\) edges.A \(\gamma\)-labeling of \(G\) is a one-to-one function f from \(V(G)\) to \({0,1,…,q}\) that induces a labeling \(f’\) from \(V(G)\) to \({1,2,…,q}\) defined by \(f(e) = |f(u) – f(v)|\) for each edge \(e = uv\) of \(G\). The value of a \(\gamma\)-labeling \(f\) is defined to be the sum of the values of \(f’\) over all
edges. Also, the maximum value of a \(\gamma\)-labeling of \(G\) is defined as the maximum of the values among all \(\gamma\)-labelings of \(G,\) while the minimum value is the minimum of the values among all \(\gamma\)-labelings
of \(G\). In this paper, the maximum value and minimum value are determined for any complete bipartite graph.