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An edge-graceful \((p, q)\)-graph \(G = (V, E)\) is a graph with \(p\) vertices and \(q\) edges for which there is a bijection \(f : E \to \{1,2,\ldots,q\}\) such that the induced mapping \(f^+ : V \to \mathbb{Z}_p\), defined by \(f^+(u) \equiv \sum\limits_{uv \in E} f(uv) \pmod{p}\), for \(u \in V\), is a bijection. In this paper, some results on edge-gracefulness of trees are extended to \(k\)-fold graphs based on graphs with \(p$ vertices and \(p – 1\) edges. A \(k\)-fold multigraph \(G[k]\) derived from a graph \(G\) is one in which each edge of \(G\) has been replaced by \(k\) parallel edges with the same vertices as the original edge. Certain classes of \(k\)-fold multigraphs derived from paths, combs, and spiders are shown to be edge-graceful, as well as other graphs constructed by combining these graphs in specified ways.