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 \(f: V(G) \to \{0, 1, 2, \ldots, m\}\) that induces a labeling \(f’: E(G) \to \{1, 2, \ldots, m\}\) of the edges of \(G\) defined by \(f'(e) = |f(u) – f(v)|\) for each edge \(e = uv\) of \(G\). The value of a \(\gamma\)-labeling \(f\) is defined as

\[val(f) = \sum\limits_{e \in E(G)} f'(e).\]

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

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

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