On graphs whose smallest eigenvalue is at least − 1 − √2

AbstractThe main result is that if the smallest eigenvalue of a graph H exceeds a fixed number larger than the smallest root (≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2, and if every vertex of H has sufficiently large valency, then the smallest eigenvalue of H is at least − 1 − √2 and the structure of H is completely characterized through a new generalization of line graphs