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

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 $En(\mathcal{S}(D))<En(\mathcal{S}(Y))$.