Abstract
For a graph G with vertex assignment c:V(G) → Z+ , we define ∑v∈V(H)c(v) 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(G) → {1, ⋯ , 12n2} such that for every n-vertex graph G' ≇ G and vertex assignment c' for G' , the corresponding collections of connected subgraph sums are different (i.e., S(G, c) ≠ S(G', c') ). We also provide some remarks on vertex assignments of a graph G for which all connected subgraph sums are different.