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The notions of sum labeling and sum graph were introduced by Harary in 1990. In a sum labeling, a vertex is called a working vertex if its label is equal to the sum of the labels of a pair of two distinct vertices.
A sum labeling of a graph \( G \) is said to be \text{exclusive} if it is a sum labeling of \( G \) such that \( G \) contains no working vertex. Any connected graph \( G \) will require some additional isolated vertices in order to be labeled exclusively. The smallest number of such isolates is called the exclusive sum number of \( G \); it is denoted by \( \epsilon(G) \). The number of isolates cannot be less than the maximum number of neighbors of any vertex in the graph, that is, at least equal to \( \Delta(G) \), the maximum vertex degree in \( G \). If \( \epsilon(G) = \Delta(G) \), then \( G \) is said to be a \( \Delta \)-\text{optimum summable graph}. An exclusive sum labeling of \( G \) using \( \Delta(G) \) isolates is called a \( \Delta \)-optimum exclusive sum labeling of \( G \).
In this paper, we show that some families of trees are \( \Delta \)-optimum summable graphs. However, this is not true for all trees, and we present an example of a tree that is not a \( \Delta \)-optimum summable graph, giving rise to an open problem.