Few Families of Harmonic Mean Graphs

A graph \(G = (V, E)\) with \(p\) vertices and \(q\) edges is called a Harmonic mean graph if it is possible to label the vertices \(v \in V\) with distinct labels \(f(v)\) from \(1, 2, \dots, q+1\) in such a way that when each edge \(e = uv\) is labeled with \(f(e = uv) = \left\lceil\frac{2f(u)f(v)}{f(u) + f(v)}\right\rceil\) or \(\left\lfloor \frac{2f(u)f(v)}{f(u) + f(v)} \right\rfloor\), then the edge labels are distinct. In this case, \(f\) is called a Harmonic mean labeling of \(G\). In this paper, we investigate some new families of Harmonic mean graphs.