On the spum and sum-diameter of paths

In a sum graph, the vertices are labeled with distinct positive integers, and two vertices are adjacent if the sum of their labels is equal to the label of another vertex. In 1990, Harary showed that not all graphs G can be labeled as a sum graph but the union of G and at least some $\sigma \left(G\right)$ isolated vertices can be. The spum of a graph G is defined as the minimum difference between the largest and smallest labels of a sum graph that consists of the union of G and exactly $\sigma \left(G\right)$ isolated vertices. More recently, Li introduced the sum-diameter of a graph G, which modifies the definition of spum by removing the requirement that the number of isolated vertices must be $\sigma \left(G\right)$. In this paper, we settle conjectures by Singla, Tiwari, and Tripathi and a conjecture by Li by evaluating the spum and the sum-diameter of paths.