Peaks on Graphs

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.