On the extremal energy of integral weighted trees

TL;DR

This paper determines the extremal energy values for weighted trees and forests with fixed total weight, characterizing the graphs that achieve these bounds, including special cases with (0,1) weights.

Contribution

It provides exact extremal energy values for weighted trees and forests with fixed total weights, including characterizations of extremal graphs and special (0,1) weight cases.

Findings

Minimum energy for weighted trees and forests identified.

Maximum energy for forests determined.

Characterizations of extremal graphs provided.

Abstract

Let ${\mathcal T}(n,m)$ and ${\mathcal F}(n,m)$ denote the classes of weighted trees and forests, respectively, of order $n$ with the positive integral weights and the fixed total weight sum $m$, respectively. In this paper, we determine the minimum energies for both the classes ${\mathcal T}(n,m)$ and ${\mathcal F}(n,m)$. We also determine the maximum energy for the class ${\mathcal F}(n,m)$. In all cases, we characterize the weighted graphs whose energies reach these extremal values. We also solve the similar maximum energy and minimum energy problems for the classes of (0,1) weighted trees and forests.