Note on extremal problems about connected subgraph sums

For a graph G with vertex assignment $c:V\left(G\right)\to Z^{+}$ , we define ${\sum }_{v\in V\left(H\right)}c\left(v\right)$ for a connected subgraph H of G as a connected subgraph sum of G. We study the set S(G, c) of connected subgraph sums and, in particular, resolve a problem posed by O.-H. S. Lo in a strong form. We show that for each n-vertex graph G, there is a vertex assignment $c:V\left(G\right)\to \{1,\cdots ,12n^2\}$ such that for every n-vertex graph $G^{\text{'}}\ncong G$ and vertex assignment $c^{\text{'}}$ for $G^{\text{'}}$ , the corresponding collections of connected subgraph sums are different (i.e., $S(G,c)\ne S(G^{\text{'}},c^{\text{'}})$ ). We also provide some remarks on vertex assignments of a graph G for which all connected subgraph sums are different.