Harmonic Mean Cordial Labeling of Some Known Graphs

All graphs considered in this paper are simple, finite, and undirected. A function f:V(G)→{1,2} is said to be a harmonic mean cordial labeling if the induced edge labeling f^*:E(G)→{1,2} defined by f^* (uv)=⌊2f(u)f(v)/(f(u)+f(v) )⌋ satisfies the conditions |v_f (1)-v_f (2)|≤1 and |e_f (1)-e_f (2)|≤1, where v_f (i) and e_f (i) denote the number of vertices and edges labeled with i, respectively. In this paper, we investigate the existence of harmonic mean cordial labeling for several classes of graphs. In particular, we prove that the jewel graph J_n with a prime edge admits harmonic mean cordial labeling if and only if n is odd, while it does not admit such labeling when n is even. Further, we show that the jewel graph without a prime edge is not harmonic mean cordial for any n∈N. We also examine the harmonic mean cordiality of graphs obtained through operations such as vertex switching and duplication in cycles, wheel graphs, bistar graphs, and fan graphs. Several new results are established, supported by appropriate constructions and counterexamples.