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

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