# Note on extremal problems about connected subgraph sums

## TL;DR

This paper solves a graph theory problem by showing how to assign values to graph vertices so that connected subgraph sums uniquely identify the graph.

## Contribution

The paper resolves a problem posed by Lo by proving a strong uniqueness result for connected subgraph sums.

## Key findings

- For any n-vertex graph G, a specific vertex assignment ensures unique connected subgraph sums.
- Non-isomorphic graphs with such assignments have different connected subgraph sum collections.
- The paper also discusses conditions for all connected subgraph sums to be distinct.

## Abstract

For a graph G with vertex assignment \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$c:V(G)\rightarrow \mathbb {Z}^+$$\end{document}c:V(G)→Z+, we define \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sum _{v\in V(H)}c(v)$$\end{document}∑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 \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$c:V(G)\rightarrow \{1,\dots ,12n^2\}$$\end{document}c:V(G)→{1,⋯,12n2} such that for every n-vertex graph \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G'\not \cong G$$\end{document}G′≇G and vertex assignment \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$c'$$\end{document}c′ for \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G'$$\end{document}G′, the corresponding collections of connected subgraph sums are different (i.e., \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S(G,c)\ne S(G',c')$$\end{document}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.