Note for Nikiforov's two conjectures on the energy of trees

Abstract

The energy $E$ of a graph is defined to be the sum of the absolute values of its eigenvalues. Nikiforov in {\it ``V. Nikiforov, The energy of $C_4$-free graphs of bounded degree, Lin. Algebra Appl. 428(2008), 2569--2573"} proposed two conjectures concerning the energy of trees with maximum degree $\Delta\leq 3$. In this short note, we show that both conjectures are true.