Abstract
The 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 x 3 + 2 x 2 − 2 x − 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.
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