Abstract
The arithmetic-geometric spectral radius of a graph G is the largest eigenvalue of the arithmetic-geometric matrix of G whose (u, v)-entry is \(\frac{d_u+d_v}{2\sqrt{d_ud_v}}\) if u and v are adjacent and 0 otherwise for \(u,v\in V(G)\), where \(d_u\) denotes the degree of vertex u in G. We determine the graphs with the largest and the next largest arithmetic-geometric spectral radii over all n-vertex bicyclic graphs with \(n\ge 5\).
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References
Brouwer AE, Haemers WH (2012) Spectra of Graphs. Springer, New York
Cui S-Y, Wang W, Tian G-X, Wu B (2021) On the arithmetic-geometric index of graphs. MATCH Commun. Math. Comput. Chem. 85:87–107
Guo X, Gao Y-B (2020) Arithmetic-geometric spectral radius and energy of graphs. MATCH Commun. Math. Comput. Chem. 83:651–660
Lin Z, Deng B, Miao L-Y, Li H (2022) On the spectral radius, energy and Estrada index of the arithmetic-geometric matrix of a graph. Discrete Math. Algorithms Appl. 14:2150108:1-2150108:20
Milovanović IŽ, Matejić MM, Milovanović EI (2018) Upper bounds for arithmetic-geometric index of graphs. Sci. Pub. State Univ. Novi Pazar A 10:49–54
Molina ED, Rodríguez JM, Sánchez JL, Sigarreta JM (2021) Some properties of the arithmetic-geometric index. Symmetry 13:857
Shegehalli VS, Kanabur RR (2015) Arithmetic-geometric indices of path graph. J. Math. Comput. Sci. 16:19–24
Vujošević S, Popivoda G, Vukićević ŽK, Furtula B, Škrekovski R (2021) Arithmetic-geometric index and its relations with geometric-arithmetic index. Appl. Math. Comput. 391:Paper 125706
Vukićević ŽK, Vujošević S, Popivoda G (2021) Unicyclic graphs with extremal values of arithmetic-geometric index. Discrete Appl. Math. 302:67–75
Wang Y-J, Gao Y-B (2020) Nordhaus-Gaddum-type relations for arithmetic-geometric spectral radius and energy. Math. Probl. Eng. 2020:Article 5898735
Zheng L, Tian G-X, Cui S-Y (2020) On spectral radius and energy of arithmetic-geometric matrix of graphs. MATCH Commun. Math. Comput. Chem. 83:635–650
Zheng L, Tian G-X, Cui S-Y (2021) Arithmetic-geometric energy of specific graphs. Discrete Math. Algorithms Appl. 13(2):Paper 2150005, 15 pp
Zheng R-L, Jin X-A, Arithmetic-geometric spectral radius of trees and unicyclic graphs, arxiv: 2105.03884
Acknowledgements
This work was supported by Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515010028).
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Communicated by Leonardo de Lima.
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Yuan, Y. On the arithmetic-geometric spectral radius of bicyclic graphs. Comp. Appl. Math. 42, 304 (2023). https://doi.org/10.1007/s40314-023-02449-w
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DOI: https://doi.org/10.1007/s40314-023-02449-w