Peaks on Graphs

TL;DR

This paper investigates the enumeration of vertex labelings in graphs where certain vertices are peaks, generalizing permutation peaks to various graph families, and provides algorithms for constructing all such labelings.

Contribution

It introduces an algorithm to construct all labelings with a specified peak set for any graph, extending the concept of peaks from permutations to broader graph classes.

Findings

Developed an algorithm for constructing labelings with a given peak set.

Extended peak set analysis to cycle graphs and joins of graphs.

Connected peak set enumeration to permutation peak studies.

Abstract

Given a graph $G$ with $n$ vertices and a bijective labeling of the vertices using the integers $1,2,\ldots, n$, we say $G$ has a peak at vertex $v$ if the degree of $v$ is greater than or equal to 2, and if the label on $v$ is larger than the label of all its neighbors. Fix an enumeration of the vertices of $G$ as $v_1,v_2,\ldots, v_{n}$ and a fix a set $S\subset V(G)$. We want to determine the number of distinct bijective labelings of the vertices of $G$, such that the vertices in $S$ are precisely the peaks of $G$. The set $S$ is called the \emph{peak set of the graph} $G$, and the set of all labelings with peak set $S$ is denoted by $\PSG$. This definition generalizes the study of peak sets of permutations, as that work is the special case of $G$ being the path graph on $n$ vertices. In this paper, we present an algorithm for constructing all of the bijective labelings in $\PSG$ for any $S\subseteq V(G)$. We also explore peak sets in certain families of graphs, including cycle graphs and joins of graphs.