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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.16810 |
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| _version_ | 1866914983708721152 |
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| author | D., Eric O. Andriantiana Sinoxolo, Xhanti |
| author_facet | D., Eric O. Andriantiana Sinoxolo, Xhanti |
| contents | 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 $σ$. 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 $En(\mathcal{S}(D))<En(\mathcal{S}(Y))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16810 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Caterpillars with given degree sequence, small Energy and small Hosoya index D., Eric O. Andriantiana Sinoxolo, Xhanti Combinatorics 05C69 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 $σ$. 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 $En(\mathcal{S}(D))<En(\mathcal{S}(Y))$. |
| title | Caterpillars with given degree sequence, small Energy and small Hosoya index |
| topic | Combinatorics 05C69 |
| url | https://arxiv.org/abs/2410.16810 |