Chemical trees minimizing energy and Hosoya index

TL;DR

This paper identifies the structure of chemical trees (max degree 4) that minimize energy and Hosoya index, proving a conjecture and showing minimal energy grows linearly with size, with explicit constants.

Contribution

It establishes that trees minimizing energy also minimize the Hosoya index in the chemical tree class, confirming a conjecture and providing explicit growth constants.

Findings

Trees with given maximum degree that minimize energy are the same as those minimizing Hosoya index.

Minimal energy grows linearly with the size of the trees.

Explicit growth constants depend only on maximum degree.

Abstract

The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph's eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. Trees minimizing the energy under various additional conditions have been determined in the past, e.g., trees with a given diameter or trees with a perfect matching. However, it is quite a natural problem to minimize the energy of trees with bounded maximum degree--clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimal energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.