Preservers of Cover Numbers and Independence Numbers of Undirected Graphs

Let \(G_n\) be the set of all simple loopless undirected graphs on \(n\) vertices. Let \(T\) be a linear mapping, \(T : G_n \rightarrow G_n\) for which the independence number of \(T(G)\) is the same as the independence number for \(G\) for any \(G \in G_n\). We show that \(T\) is necessarily a vertex permutation. Similar results are obtained for mappings preserving the matching number of bipartite graphs, the vertex cover number of undirected graphs, and the edge independence number of undirected graphs.