Spectral moments of trees with given degree sequence

Abstract

Let λ1,…,λn be the eigenvalues of a graph G. For any k⩾0, the k-th spectral moment of G is defined by Mk(G)=λ1k+⋯+λnk. We use the fact that Mk(G) is also the number of closed walks of length k in G to show that among trees T whose degree sequence is D or majorized by D, Mk(T) is maximized by the greedy tree with degree sequence D (constructed by assigning the highest degree in D to the root, the second-, third-, …highest degrees to the neighbors of the root, and so on) for any k⩾0. Several corollaries follow, in particular a conjecture of Ilić and Stevanović on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Marković and Glišić on the Estrada index of such trees, which is defined as EE(G)=eλ1+⋯+eλn.