Editing Spectral graph theory (section)

=== Finding cospectral graphs ===
[[Almost all]] [[tree (graph theory)|tree]]s are cospectral, i.e., as the number of vertices grows, the fraction of trees for which there exists a cospectral tree goes to 1.{{sfnp|Schwenk|1973|pages =275-307}}

A pair of [[regular graph]]s are cospectral if and only if their complements are cospectral.<ref>{{Cite web|url=http://www.math.uwaterloo.ca/~cgodsil/pdfs/cospectral.pdf|title=Are Almost All Graphs Cospectral?|last=Godsil|first=Chris|date=November 7, 2007}}</ref>

A pair of [[distance-regular graph]]s are cospectral if and only if they have the same intersection array.

Cospectral graphs can also be constructed by means of the [[isospectral|Sunada method]].<ref>{{citation
 | last = Sunada | first = Toshikazu | journal = Ann. of Math.
 | pages = 169–186
 | title = Riemannian coverings and isospectral manifolds
 | volume = 121
 | issue = 1 | year = 1985
 | doi = 10.2307/1971195 | jstor = 1971195 }}.</ref>

Another important source of cospectral graphs are the point-collinearity graphs and the line-intersection graphs of [[incidence geometry|point-line geometries]]. These graphs are always cospectral but are often non-isomorphic.<ref>{{harvnb|Brouwer|Haemers|2011}}</ref>