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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2410.09830 |
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| _version_ | 1866912315230650368 |
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| author | Tang, Quanyu Liu, Yinchen Wang, Wei |
| author_facet | Tang, Quanyu Liu, Yinchen Wang, Wei |
| contents | In this paper, we introduce the concepts of positive and negative $p$-energies of graphs and investigate their behavior under edge addition. Specifically, we generalize the classical notions of positive and negative square energies to the $p$-energy setting, denoted by $\mathcal{E}_p^{+}(G)$ and $\mathcal{E}_p^{-}(G)$, respectively. We establish improved lower bounds for these quantities under edge addition, which sharpen existing results by Abiad et al.\ in the case $p=2$. Furthermore, we address the monotonicity problem for $\mathcal{E}_p^{+}(G)$ under edge addition, and construct a family of counterexamples showing that monotonicity fails for $1 \leq p < 3$. Finally, we conclude with several open problems for further investigation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09830 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Positive and Negative $p$-Energies of Graphs under Edge Addition Tang, Quanyu Liu, Yinchen Wang, Wei Combinatorics 05C50 In this paper, we introduce the concepts of positive and negative $p$-energies of graphs and investigate their behavior under edge addition. Specifically, we generalize the classical notions of positive and negative square energies to the $p$-energy setting, denoted by $\mathcal{E}_p^{+}(G)$ and $\mathcal{E}_p^{-}(G)$, respectively. We establish improved lower bounds for these quantities under edge addition, which sharpen existing results by Abiad et al.\ in the case $p=2$. Furthermore, we address the monotonicity problem for $\mathcal{E}_p^{+}(G)$ under edge addition, and construct a family of counterexamples showing that monotonicity fails for $1 \leq p < 3$. Finally, we conclude with several open problems for further investigation. |
| title | On the Positive and Negative $p$-Energies of Graphs under Edge Addition |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2410.09830 |