On a conjecture of Nikiforov concerning the minimal $p$-energy of connected graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866910868947599360 |
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| author | Tang, Quanyu Liu, Yinchen Wang, Wei |
| author_facet | Tang, Quanyu Liu, Yinchen Wang, Wei |
| contents | For a given simple graph \( G \), the \( p \)-energy of \( G \), denoted by \( \mathcal{E}_p(G) \), is defined as the sum of the \( p \)-th power of the absolute values of the eigenvalues of its adjacency matrix. Let \( S_n \) denote the star graph with one internal node and \( n-1 \) leaves. Nikiforov conjectured that for \( 1 < p < 2 \), the connected graph of order \( n \) with the smallest \( p \)-energy is \( S_n \). Recently, this conjecture was proved for bipartite graphs. In this paper, by employing a Coulson-Jacobs-type formula and certain spectral radius results for connected graphs, we completely resolve this conjecture. Furthermore, we establish that the equality condition in the inequality \( \mathcal{E}_p(G) \geq \mathcal{E}_p(S_n) \) holds if and only if \( G \) is \( S_n \). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a conjecture of Nikiforov concerning the minimal $p$-energy of connected graphs Tang, Quanyu Liu, Yinchen Wang, Wei Combinatorics 05C50 For a given simple graph \( G \), the \( p \)-energy of \( G \), denoted by \( \mathcal{E}_p(G) \), is defined as the sum of the \( p \)-th power of the absolute values of the eigenvalues of its adjacency matrix. Let \( S_n \) denote the star graph with one internal node and \( n-1 \) leaves. Nikiforov conjectured that for \( 1 < p < 2 \), the connected graph of order \( n \) with the smallest \( p \)-energy is \( S_n \). Recently, this conjecture was proved for bipartite graphs. In this paper, by employing a Coulson-Jacobs-type formula and certain spectral radius results for connected graphs, we completely resolve this conjecture. Furthermore, we establish that the equality condition in the inequality \( \mathcal{E}_p(G) \geq \mathcal{E}_p(S_n) \) holds if and only if \( G \) is \( S_n \). |
| title | On a conjecture of Nikiforov concerning the minimal $p$-energy of connected graphs |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2410.16604 |