Cospectral regular graphs with and without a perfect matching
  For each $b \geq 5$ we construct a pair of cospectral $b$-regular graphs, where one has a perfect matching and the other one not. This solves a research problem posed by the third author at the 22nd British Combinatorial Conference. 
Zoltan L. Blazsik, Jay Cummings, Willem H. Haemers
math.CO