An upper bound for difference of energies of a graph and its complement
The A-energy of a graph G, denoted by EA(G), is defined as sum of the absolute values of eigenvalues of adjacency matrix of G. Nikiforov in Nikiforov (2016) proved that EA(G¯)−EA(G)≤2μ¯1and EA(G)−EA(G¯)≤2μ1for any graph G and posed a problem to find best possible upper bound for EA(G)−EA(G¯), where...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Elsevier
2023-11-01
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| Series: | Examples and Counterexamples |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666657X23000022 |
| _version_ | 1797796355961257984 |
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| author | Harishchandra S. Ramane B. Parvathalu K. Ashoka |
| author_facet | Harishchandra S. Ramane B. Parvathalu K. Ashoka |
| author_sort | Harishchandra S. Ramane |
| collection | DOAJ |
| description | The A-energy of a graph G, denoted by EA(G), is defined as sum of the absolute values of eigenvalues of adjacency matrix of G. Nikiforov in Nikiforov (2016) proved that EA(G¯)−EA(G)≤2μ¯1and EA(G)−EA(G¯)≤2μ1for any graph G and posed a problem to find best possible upper bound for EA(G)−EA(G¯), where μ1and μ1¯are the largest adjacency eigenvalues of G and its complement G¯respectively. We attempt to provide an answer by giving an improved upper bound on a class of graphs where regular graphs become particular case. As a consequence, it is proved that there is no strongly regular graph with negative eigenvalues greater than −1. The obtained results also improves some of the other existing results. |
| first_indexed | 2024-03-13T03:31:51Z |
| format | Article |
| id | doaj.art-df0b09eb725d42aebf49348155068c29 |
| institution | Directory Open Access Journal |
| issn | 2666-657X |
| language | English |
| last_indexed | 2024-03-13T03:31:51Z |
| publishDate | 2023-11-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Examples and Counterexamples |
| spelling | doaj.art-df0b09eb725d42aebf49348155068c292023-06-24T05:19:35ZengElsevierExamples and Counterexamples2666-657X2023-11-013100100An upper bound for difference of energies of a graph and its complementHarishchandra S. Ramane0B. Parvathalu1K. Ashoka2Department of Mathematics, Karnatak University, Dharwad 580003, India; Corresponding author.Department of Mathematics, Karnatak University’s Karnatak Arts College, Dharwad 580001, IndiaPG Department of Mathematics, Karnatak University’s Karnatak Science College, Dharwad 580001, IndiaThe A-energy of a graph G, denoted by EA(G), is defined as sum of the absolute values of eigenvalues of adjacency matrix of G. Nikiforov in Nikiforov (2016) proved that EA(G¯)−EA(G)≤2μ¯1and EA(G)−EA(G¯)≤2μ1for any graph G and posed a problem to find best possible upper bound for EA(G)−EA(G¯), where μ1and μ1¯are the largest adjacency eigenvalues of G and its complement G¯respectively. We attempt to provide an answer by giving an improved upper bound on a class of graphs where regular graphs become particular case. As a consequence, it is proved that there is no strongly regular graph with negative eigenvalues greater than −1. The obtained results also improves some of the other existing results.http://www.sciencedirect.com/science/article/pii/S2666657X23000022Graph energyStrongly regular graphEquienergetic graphs |
| spellingShingle | Harishchandra S. Ramane B. Parvathalu K. Ashoka An upper bound for difference of energies of a graph and its complement Examples and Counterexamples Graph energy Strongly regular graph Equienergetic graphs |
| title | An upper bound for difference of energies of a graph and its complement |
| title_full | An upper bound for difference of energies of a graph and its complement |
| title_fullStr | An upper bound for difference of energies of a graph and its complement |
| title_full_unstemmed | An upper bound for difference of energies of a graph and its complement |
| title_short | An upper bound for difference of energies of a graph and its complement |
| title_sort | upper bound for difference of energies of a graph and its complement |
| topic | Graph energy Strongly regular graph Equienergetic graphs |
| url | http://www.sciencedirect.com/science/article/pii/S2666657X23000022 |
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