Convex combination of first and second eigenvalues of trees

Abstract

For a graph $G$, let $λ_1(G)$ and $λ_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum $λ_1(G) + λ_2(G)$ is called the \emph{spectral sum} of $G$. We investigate the spectral sum of trees of order $n$ and determine the extremal trees that attain the maximum/minimum. Moreover, for any $α\in [0,1],$ we describe the extremal trees which maximize the convex combination $αλ_1 + (1-α)λ_2$ in the class of $n$-vertex trees for sufficiently large $n$.