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 |
| Summary: | 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. |
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| ISSN: | 2666-657X |