Por Ali Mohammadian (School of Mathematics, Institute for Research in Fundamental Sciences - IPM, Tehran, Iran).

Abstract: A graph G is called integral if all eigenvalues of its adjacency matrix, A(G), consist entirely of integers. The nullity of G is the nullity of A(G), that is the multiplicity of 0 as an eigenvalue of A(G). In this talk, we are concerned with integral trees. These objects are extremely rare and very difficult to find. We first present a short survey on integral graphs. We show that for any integer d >1, there are infinitely many integral trees of diameter d. We will also show that for any integer k >1, there are only finitely many integral trees with nullity k.