The Neighbour Sum Problem on Trees

TL;DR

This paper investigates the neighbour sum property in trees, providing an algorithm to verify this property and identifying a broad class of trees that satisfy it, advancing understanding of graph labelings.

Contribution

It introduces an algorithm to determine if a finite tree satisfies the neighbour sum property and characterizes a large class of such trees.

Findings

Algorithm for checking neighbour sum property in trees

Identification of a large class of trees satisfying the property

Enhanced understanding of neighbour sum labelings in trees

Abstract

A graph $\mathcal G = (\mathcal V, \mathcal E)$ is said to satisfy the Neighbour Sum Property if there exists some $f:\mathcal V\to\mathbb R$ such that $f\not\equiv 0$ and it maps every vertex to the sum of the values taken by its neighbours. In this article, we provide an algorithm to check whether a given finite tree satisfies the neighbour sum property. We also find a large class of trees on $n$ vertices that satisfy the property.