Abstract

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 $x\in V$ with distinct labels $f(x)$ from $1,2,\ldots,q+1$ in such a way that when each edge $e=uv$ is labeled with $f(uv) = \left\lceil \frac{2f(u)f(v)}{f(u)+f(v)}\right\rceil$ 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 Harmonic mean labeling of $G$.

The concept of Harmonic mean labeling was introduced in (Somasundaram, Ponraj \& Sandhya). In (Somasundaram, Ponraj \& Sandhya) and (Sandhya, Somasundaram \& Ponraj, 2012) we investigate the harmonic mean labeling of several standard graphs such as path, cycle comb, ladder, Triangular snakes, Quadrilateral snakes etc. In the present paper, we investigate the harmonic mean labeling for a polygonal chain, square of the path and dragon. Also we enumerate all harmonic mean graph of order $\leq5$.

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