Linear Operators on Graphs which Preserve the Dot-Product Dimension

Abstract

Let ${\mathcal{G}}_n$ be the set of all simple loopless undirected graphs on $n$ vertices. Let $T$ be a linear mapping, $T:{\mathcal{G}}_n\to {\mathcal{G}}_n$, such that the dot product dimension of $T(G)$ is the same as the dot product dimension of $G$ for any $G\in {\mathcal{G}}_n$. We show that $T$ is necessarily a vertex permutation. Similar results are obtained for mappings that preserve sets of graphs with specified dot product dimensions.