On the limit points of the smallest eigenvalues of regular graphs

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval ${-1-\sqrt2, -2)}$ and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval ${\alpha_1, -1-\sqrt2)}$ where α ₁ is the smallest root ${(\approx -2.4812)}$ of the polynomial x ³?+?2x ² ? 2x ? 2. From these results, we determine the largest and second largest limit points of smallest eigenvalues of regular graphs less than ?2. Moreover we determine the supremum of the smallest eigenvalue among all connected 3-regular graphs with smallest eigenvalue less than ?2 and we give the unique graph with this supremum value as its smallest eigenvalue.