On the $\mathcal{ABS}$ spectrum and energy of graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916382933778432 |
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| author | Shetty, Swathi Rakshith, B. R. N. V, Sayinath Udupa |
| author_facet | Shetty, Swathi Rakshith, B. R. N. V, Sayinath Udupa |
| contents | Let $η_{1}\ge η_{2}\ge\cdots\ge η_{n}$ be the eigenavalues of $\mathcal{ABS}$ matrix. In this paper, we characterize connected graphs with $\mathcal{ABS}$ eigenvalue $η_{n}>-1$. As a result, we determine all connected graphs with exactly two distinct $\mathcal{ABS}$ eigenvalues. We show that a connected bipartite graph has three distinct $\mathcal{ABS}$ eigenvalues if and only if it is a complete bipartite graph. Furthermore, we present some bounds for the $\mathcal{ABS}$ spectral radius (resp. $\mathcal{ABS}$ energy) and characterize extremal graphs. Also, we obtain a relation between $\mathcal{ABC}$ energy and $\mathcal{ABS}$ energy. Finally, the chemical importance of $\mathcal{ABS}$ energy is investigated and it shown that the $\mathcal{ABS}$ energy is useful in predicting certain properties of molecules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $\mathcal{ABS}$ spectrum and energy of graphs Shetty, Swathi Rakshith, B. R. N. V, Sayinath Udupa Combinatorics 05C50, 05C09, 05C35, 05C92 Let $η_{1}\ge η_{2}\ge\cdots\ge η_{n}$ be the eigenavalues of $\mathcal{ABS}$ matrix. In this paper, we characterize connected graphs with $\mathcal{ABS}$ eigenvalue $η_{n}>-1$. As a result, we determine all connected graphs with exactly two distinct $\mathcal{ABS}$ eigenvalues. We show that a connected bipartite graph has three distinct $\mathcal{ABS}$ eigenvalues if and only if it is a complete bipartite graph. Furthermore, we present some bounds for the $\mathcal{ABS}$ spectral radius (resp. $\mathcal{ABS}$ energy) and characterize extremal graphs. Also, we obtain a relation between $\mathcal{ABC}$ energy and $\mathcal{ABS}$ energy. Finally, the chemical importance of $\mathcal{ABS}$ energy is investigated and it shown that the $\mathcal{ABS}$ energy is useful in predicting certain properties of molecules. |
| title | On the $\mathcal{ABS}$ spectrum and energy of graphs |
| topic | Combinatorics 05C50, 05C09, 05C35, 05C92 |
| url | https://arxiv.org/abs/2409.03287 |