On graphs with the smallest eigenvalue at least -1- square root 2, part III - Kubota, Sho (Ars mathematica contemporanea, 2019)
There are many results on graphs with the smallest eigenvalue at least ?$-2$?. In order to study graphs with the eigenvalues at least ?$-1 - \sqrt{2}$?, R. Woo and A. Neumaie Linear Algebra Appl. 226-228, 577-591 (1995) introduced Hoffman graphs and ?$\mathcal{H}$?-line graphs. They proved that a graph with the sufficiently large minimum degree and the smallest eigenvalue at least ?$-1 - \sqrt{2}$? is a slim ?$\{\mathfrak h_2, \mathfrak h_5, \mathfrak h_7, \mathfrak h_9\}$?-line graph. After that, the second author Ars Math. Contemp. 5, No. 2, 243--258 (2012) researched on slim ?$\{\mathfrak h_2, \mathfrak h_5\}$?-line graphs. As an analogue, we reveal the condition under which a strict ?$\{\mathfrak h_1, \mathfrak h_4, \mathfrak h_7\}$?-cover of a slim ?$\{\mathfrak h_7\}$?-line graph is unique, and completely determine the minimal forbidden graphs for the slim ?$\{\mathfrak h_7\}$?-line graphs.
Keywords: Hoffman graph; Hoffmanov graf; line graph; najmanjša lastna vrednost; povezavni graf; smallest eigenvalue