Convex combination of first and second eigenvalues of trees

Abstract

For a graph $G$, let ${\lambda }_1(G)$ and ${\lambda }_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum ${\lambda }_1(G)+{\lambda }_2(G)$ 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 $\alpha \in [0,1],$ we describe the extremal trees which maximize the convex combination $\alpha {\lambda }_1+(1-\alpha ){\lambda }_2$ in the class of $n$-vertex trees for sufficiently large $n$.

Improved version; some errors corrected