On the extremal energy of integral weighted graphs

Abstract

Let 𝒯(n, m) and 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 article, we determine the minimum energies for both the classes 𝒯(n, m) and ℱ(n, m). We also determine the maximum energy for the class 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.

Keywords:

• energy
• graph
• weighted graph
• integral weights
• tree
• forest

AMS Subject Classifications::

• 05C50
• 05D99
• 15A18

Acknowledgements

The second author's research was supported by the National Science Foundation of China (10731040), the fourth author's research by the National Science Foundation of China (10871230), and the last author's research by the Zhejiang Provincial Natural Science Foundation of China (Y7080364).