Seidel matrices, Dilworth number and an eigenvalue-free interval for cographs

A graph G=(VG,EG) is said to be a cograph if the path P4 does not appear among its induced subgraphs. The vicinal preorder ≺ on the vertex set VG is defined in terms of inclusions between neighborhoods. The minimum number ∇(G) of ≺-chains required to cover G is called the Dilworth number of G. In this paper it is proved that for a cograph G, the multiplicity of every Seidel eigenvalue λ≠±1 does not exceed ∇(G). This bound turns out to be tight and can be further improved for threshold graphs. Moreover, it is shown that cographs with at least two vertices have no Seidel eigenvalues in the interval (−1,1).