Energy of strong reciprocal graphs
The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute values of all eigenvalues of $G$. A graph $G$ is called reciprocal if $ \frac{1}{\lambda} $ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $ \lambda $ and $\frac{1}{\lambda}$ ha...
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University of Isfahan
2023-09-01
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| Series: | Transactions on Combinatorics |
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| Online Access: | https://toc.ui.ac.ir/article_26810_877d7be13cf3ac469149b724447166ab.pdf |
| _version_ | 1811344270156627968 |
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| author | Maryam Ghahremani Abolfazl Tehranian Hamid Rasouli Mohammad Ali Hosseinzadeh |
| author_facet | Maryam Ghahremani Abolfazl Tehranian Hamid Rasouli Mohammad Ali Hosseinzadeh |
| author_sort | Maryam Ghahremani |
| collection | DOAJ |
| description | The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute values of all eigenvalues of $G$. A graph $G$ is called reciprocal if $ \frac{1}{\lambda} $ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $ \lambda $ and $\frac{1}{\lambda}$ have the same multiplicities, for each eigenvalue $\lambda$, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631--633), it was conjectured that for every graph $G$ with maximum degree $\Delta(G)$ and minimum degree $\delta(G)$ whose adjacency matrix is non-singular, $\mathcal{E}(G) \geq \Delta(G) + \delta(G)$ and the equality holds if and only if $G$ is a complete graph. Here, we prove the validity of this conjecture for some strong reciprocal graphs. Moreover, we show that if $G$ is a strong reciprocal graph, then $\mathcal{E}(G) \geq \Delta(G) + \delta(G) - \frac{1}{2}$. Recently, it has been proved that if $G$ is a reciprocal graph of order $n$ and its spectral radius, $\rho$, is at least $4\lambda_{min}$, where $ \lambda_{min}$ is the smallest absolute value of eigenvalues of $G$, then $\mathcal{E}(G) \geq n+\frac{1}{2}$. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption. |
| first_indexed | 2024-04-13T19:45:52Z |
| format | Article |
| id | doaj.art-daae52906dc94cf3ab99977a7c43fa9e |
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| issn | 2251-8657 2251-8665 |
| language | English |
| last_indexed | 2024-04-13T19:45:52Z |
| publishDate | 2023-09-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | Transactions on Combinatorics |
| spelling | doaj.art-daae52906dc94cf3ab99977a7c43fa9e2022-12-22T02:32:45ZengUniversity of IsfahanTransactions on Combinatorics2251-86572251-86652023-09-0112316517110.22108/toc.2022.134259.199926810Energy of strong reciprocal graphsMaryam Ghahremani0Abolfazl Tehranian1Hamid Rasouli2Mohammad Ali Hosseinzadeh3Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranScience and Research Branch, Islamic Azad UniversityDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranFaculty of Engineering Modern Technologies, Amol University of Special Modern Technologies, Amol, IranThe energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute values of all eigenvalues of $G$. A graph $G$ is called reciprocal if $ \frac{1}{\lambda} $ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $ \lambda $ and $\frac{1}{\lambda}$ have the same multiplicities, for each eigenvalue $\lambda$, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631--633), it was conjectured that for every graph $G$ with maximum degree $\Delta(G)$ and minimum degree $\delta(G)$ whose adjacency matrix is non-singular, $\mathcal{E}(G) \geq \Delta(G) + \delta(G)$ and the equality holds if and only if $G$ is a complete graph. Here, we prove the validity of this conjecture for some strong reciprocal graphs. Moreover, we show that if $G$ is a strong reciprocal graph, then $\mathcal{E}(G) \geq \Delta(G) + \delta(G) - \frac{1}{2}$. Recently, it has been proved that if $G$ is a reciprocal graph of order $n$ and its spectral radius, $\rho$, is at least $4\lambda_{min}$, where $ \lambda_{min}$ is the smallest absolute value of eigenvalues of $G$, then $\mathcal{E}(G) \geq n+\frac{1}{2}$. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption.https://toc.ui.ac.ir/article_26810_877d7be13cf3ac469149b724447166ab.pdfgraph energystrong reciprocal graphnon-singular graph |
| spellingShingle | Maryam Ghahremani Abolfazl Tehranian Hamid Rasouli Mohammad Ali Hosseinzadeh Energy of strong reciprocal graphs Transactions on Combinatorics graph energy strong reciprocal graph non-singular graph |
| title | Energy of strong reciprocal graphs |
| title_full | Energy of strong reciprocal graphs |
| title_fullStr | Energy of strong reciprocal graphs |
| title_full_unstemmed | Energy of strong reciprocal graphs |
| title_short | Energy of strong reciprocal graphs |
| title_sort | energy of strong reciprocal graphs |
| topic | graph energy strong reciprocal graph non-singular graph |
| url | https://toc.ui.ac.ir/article_26810_877d7be13cf3ac469149b724447166ab.pdf |
| work_keys_str_mv | AT maryamghahremani energyofstrongreciprocalgraphs AT abolfazltehranian energyofstrongreciprocalgraphs AT hamidrasouli energyofstrongreciprocalgraphs AT mohammadalihosseinzadeh energyofstrongreciprocalgraphs |