The $\gamma$-Spectrum of a Graph

Abstract

Let $G$ be a graph of order $n$ and size $m$. A $\gamma$-labeling of $G$ is a one-to-one function $f:V(G)\to \{0,1,2,\dots ,m\}$ that induces a labeling $f^{\prime }:E(G)\to \{1,2,\dots ,m\}$ of the edges of $G$ defined by $f^{\prime }(e)=|f(u)–f(v)|$ for each edge $e=uv$ of $G$. The value of a $\gamma$-labeling $f$ is defined as

$$val(f)=\sum _{e\in E(G)}{f}^{\prime }(e).$$

The $\gamma$-spectrum of a graph $G$ is defined as

$$spec(G)=\{val(f):f\text{ is a }\gamma \text{-labeling of }G\}.$$

The $\gamma$-spectra of paths, cycles, and complete graphs are determined.