Linear Operators on Graphs which Preserve the Dot-Product Dimension

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.