Note on extremal problems about connected subgraph sums

For a graph G with vertex assignment c:V(G)→ Z⁺ c : V ( G ) → Z + , we define ∑ v∈ V(H)c(v) ∑ 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\ c : V ( G ) → { 1 , ⋯ , 12 n 2 } such that for every n -vertex graph G' G G ′ ≇ G and vertex assignment $$c'$$ c ′ for $$G'$$ G ′ , the corresponding collections of connected subgraph sums are different (i.e., S(G,c)≠ S(G',c') 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.