Behruz Tayfeh-Rezaie: Integral graphs and trees

Abstract: A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Integral graphs are very rare and difficult to find. We first introduce some general methods for constructing such graphs. As a consequence, some infinite families of integral graphs are obtained. We then pay our attention to trees where we discuss the recent construction of integral trees with arbitrary large diameter. Finally, we prove that for a given nullity more than 1, there are only finitely many integral trees.