Some results on harmonic mean graphs (Q2907681)

A graph \(G\) is called a Harmonic mean graph if it is possible to label its vertices with distinct labels \(f(x)\in\{1,2,\dots,|E(G)|+1\}\) in such a way that when each edge \(e=uv\) is labelled with NEWLINE\[NEWLINEf(uv)=\left\lceil\frac{2f(u)f(v)}{f(u)+f(v)}\right\rceil \text{or} \left\lfloor\frac{2f(u)f(v)}{f(u)+f(v)}\right\rfloorNEWLINE\]NEWLINE then the edge labels are distinct.NEWLINENEWLINEThe main contribution of the paper is a constructive proof that selected classes of graphs, namely \(C_m\cup P_n\), \(m\geq 3\), \(n>1\), \(C_m \cup C_n\), \(m\geq 3\), \(n\geq3\), \(nK_3\), \(nK_3 \cup P_m\), \(m>1\), \(mC_4\), \(mC_4\cup P_n\), \(nK_3\cup mC_4\) and crown are harmonic mean graphs.