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

TL;DR

This paper proves two conjectures by Nikiforov regarding the maximum energy of trees with degree at most 3, confirming their validity.

Contribution

It provides a proof for two previously unconfirmed conjectures about the energy bounds of certain trees, advancing understanding in spectral graph theory.

Findings

Both conjectures are confirmed as true.

The energy bounds for trees with maximum degree 3 are established.

The results contribute to spectral graph theory and graph energy studies.

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.