The Overgraphs of Generalized Cospectral Controllable Graphs
Two graphs are said to be generalized cospectral if they have the same characteristic polynomials and so do their complements. A graph is controllable if its walk matrix is nonsingular; equivalently, if all the eigenvalues of its adjacency matrix are simple and main. A graph $H$ on $(n+1)$ vertices is an overgraph of another graph $G$ on $n$ vertices if $G$ is a vertex-deleted subgraph of $H$. We prove that no two distinct overgraphs of a controllable graph are generalized cospectral; this strengthens an earlier result that stated that no two such overgraphs are isomorphic. Moreover, we present methods that produce pairs of generalized cospectral graphs $G^\prime$ and $H^\prime$ starting from a pair of generalized cospectral, non-isomorphic, controllable graphs $G$ and $H$. We show that if $G^\prime$ and $H^\prime$ are controllable, then they are non-isomorphic.