Understanding the properties of isospectral points and pairs in graphs: The concept of orthogonal relation

The mathematical property orthogonal relationship is used in proving the fact that isospectrality, isocodality and isocoefficiency of vertices within a graph are all equivalent. The same is true for isospectrality, strict isocodality and strict isocoefficiency of pairs (including edges) within a graph, whereas the weak versions of the latter properties are necessary but not sufficient for isospectrality of pairs. Similarly, necessary and sufficient conditions for isospectrality of vertices and pairs in different graphs are derived. In all these proofs, the concept of orthogonal relation plays a major role in that it allows the use of tools of elementary linear algebra.