Small energy and Hosoya index among caterpillars with a given degree sequence

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. We also show that \(Z(\mathcal {S}(D))<Z(\mathcal {S}(Y))\) and \(En(\mathcal {S}(D))<En(\mathcal {S}(Y))\) for any degree sequences \(Y=(y_1,\dots ,y_n)\) and \(D=(d_1,\dots ,d_n)\) with

$$\sum _{i=1}^{n}y_i=\sum _{i=1}^{n}d_i\text { and }\sum _{i=1}^{k}y_i\le \sum _{i=1}^{k}d_i \text { for all }1\le k \le n.$$