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Let \( G = (V, E) \) be a simple, finite, and undirected graph. A sum labeling is a one-to-one mapping \( L \) from a set of vertices of \( G \) to a finite set of positive integers \( S \) such that if \( u \) and \( v \) are vertices of \( G \), then \( uv \) is an edge in \( G \) if and only if there is a vertex \( w \) in \( G \) and \( L(w) = L(u) + L(v) \). A graph \( G \) that has a sum labeling is called a sum graph. The minimal isolated vertex that is needed to make \( G \) a sum labeling is called the sum number of \( G \), denoted as \( \sigma(G) \). The sum number of a sum graph \( G \) is always greater than or equal to \( \delta(G) \), the minimum degree of \( G \). An optimum sum graph is a sum graph that has \( \sigma(G) = \delta(G) \). In this paper, we discuss sum numbers of finite unions of some families of optimum sum graphs, such as cycles and friendship graphs.