Periodica Mathematica Hungarica Vol. 3 (1--2), (1973), pp. 175--182. ON THE EIGENVALUES OF TREES by L. LOVASZ and J. PELIKAN (Budapest) To the memory o] A. 1~I~:6-u Given a graph G (without loops and multiple edges) of n vertices labelled by 1, 2 ..... n, we can form the adjacency matrix As = (aij) of G, defined by [1 if the i th and jt h vertices are joined by an edge, aiJ = [0 otherwise. The adjacency matrix depends on the labelling of the vertices but its characteristic equation (and, consequently, its eigenvalues too) depend only on the graph G itself. As Aa is a symmetric matrix, these eigenvalues, called the eigenvalues of G, are real. We denote by fG(2) the characteristic polynomial det (2I- A~) of A~ and by A(G) its largest root. We shall begin with several general remarks on f~(2) and A(G), used in latter considerations. These propositions are special cases or easy consequen- ces of general theorems on eigenvalues of non-negative matrices (see, e. g. [2] and [3]). Although they may be well-known for the reader, it may have some use to list them here. Our main concern in this paper will be fa(2) and A(G) in the case when G is a tree (or more generally, a forest). We determine the maximal and minimal value of A(G) among all trees of n vertices and give a method which enables us to determine the order of largest eigenvalues of two different trees in several cases. NOTATIONS. V(G) and E(G) are the sets of vertices and edges of G, respectively. G ~ G' means that G and G' are isomorphic. If G 1 and G2 are rf.~ arbitrary graphs then G 1 + G 2 is defined as follows: we consider a G1 ~ G1 and a G~ ~ G2 such that V(G~) [7 V(G~) = ~ and let V(G 1 ~- G2) = V(G~) zr- V(G~), E(G 1 -~ G2) = E(G~) ~ E(G~). G1 ~ G2 is uniquely determined up to isomorphism. If eEE{G) and xEV(G) then G--e,G--x,G-- [e] denote the graphs arising from G by the removal of the edge e, of the vertex x and of the endpoints of e, respectively. If e = (x,y) is a non-adjacent pair of vertices of G then G (J e denotes the graph obtained by adding the edge e to G. G' _c G means that V(G') = V(G), E(G') c B(G).