Caterpillars with given degree sequence, small Energy and small Hosoya index

TL;DR

This paper characterizes caterpillar graphs with a given degree sequence that minimize energy and Hosoya index, and compares these properties across majorized degree sequences, revealing inequalities between them.

Contribution

It provides a characterization of caterpillars with minimal energy and Hosoya index for any given degree sequence and compares these properties for majorized sequences.

Findings

Caterpillars with alternating large and small degrees minimize energy and Hosoya index.

For majorized degree sequences, the minimal energy and Hosoya index increase.

The paper establishes inequalities relating these graph invariants across majorized sequences.

Abstract

The energy $En(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. The Hosoya index $Z(G)$ of a graph $G$ is the number of independent edge subsets of $G$, including the empty set. For any given degree sequence $D$, we characterize the caterpillar $\mathcal{S}(D)$ that has the minimum $Z$ and $En$. %and maximum $\sigma$. In $\mathcal{S}(D)$, as we move along the internal path towards the center, large and small degrees alternate. We also compare $\mathcal{S}(D)$ with $\mathcal{S}(Y)$, for a degree sequence $Y$ majorized by a degree sequence $D$. Suppose $Y=(y_1,\dots ,y_n)$ and $D=(d_1,\dots ,d_n)$ are degree sequences such that $Y$ is majorized by $D$ and$$\sum_{i=1}^{n}y_i=\sum_{i=1}^{n}d_i,$$then $Z(\mathcal{S}(D))<Z(\mathcal{S}(Y))$ and $En(\mathcal{S}(D))<En(\mathcal{S}(Y))$.