A series of trees with the first $\lfloor\frac{n-7}{2}\rfloor$ largest energies

TL;DR

This paper introduces a new method for comparing graph energies, identifies the top energy trees of certain sizes, and simplifies a proof related to maximal energy trees, advancing understanding in spectral graph theory.

Contribution

It presents a novel comparison method for energies of bipartite graphs and determines the largest energy trees, also providing a simplified proof of a conjecture on maximal energy trees.

Findings

Identified the first rac{n-7}{2} largest energy trees for n  31.

Developed a new method to compare energies of k-subdivision bipartite graphs.

Provided a simplified proof of the conjecture on the fourth maximal energy tree.

Abstract

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph. In this paper, we present a new method to compare the energies of two $k$-subdivision bipartite graphs on some cut edges. As the applications of this new method, we determine the first $\lfloor\frac{n-7}{2}\rfloor$ largest energy trees of order $n$ for $n\ge 31$, and we also give a simplified proof of the conjecture on the fourth maximal energy tree.