Note on extremal problems about connected subgraph sums

TL;DR

This paper investigates extremal problems related to sums of vertex weights over connected subgraphs, providing a construction that uniquely identifies a graph among all others via these sums.

Contribution

It proves that for any n-vertex graph, there exists a vertex assignment that uniquely determines the graph by its connected subgraph sums, resolving a problem posed by Solomon Lo.

Findings

Existence of a vertex assignment distinguishing any n-vertex graph from all others.

Construction of vertex assignments with bounded weights (up to 12n^2).

Remarks on vertex assignments where all connected subgraph sums are distinct.

Abstract

For a graph $G$ with vertex assignment $c:V(G)\to \mathbb{Z}^+$, we define $\sum_{v\in V(H)}c(v)$ for $H$ a connected subgraph 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 Solomon Lo in a strong form. We show that for each $n$-vertex graph, there is a vertex assignment $c:V(G)\to \{1,\dots,12n^2\}$ such that for every $n$-vertex graph $G'\not\cong G$ and vertex assignment $c'$ for $G'$, the corresponding collections of connected subgraph sums are different (i.e., $S(G,c)\neq S(G',c')$). We also provide some remarks on vertex assignments of a graph $G$ for which all connected subgraph sums are different.