Note on extremal problems about connected subgraph sums

Abstract For a graph G with vertex assignment $$c:V(G)\rightarrow \mathbb {Z}^+$$ c : V ( G ) → Z + , we define $$\sum _{v\in V(H)}c(v)$$ ∑ 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 $$c:V(G)\rightarrow \{1,\dots ,12n^2\}$$ c : V ( G ) → { 1 , ⋯ , 12 n 2 } such that for every n -vertex graph $$G'\not \cong G$$ G ′ ≇ G and vertex assignment $$c'$$ c ′ for $$G'$$ G ′ , the corresponding collections of connected subgraph sums are different (i.e., $$S(G,c)\ne S(G',c')$$ 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.