The \(A_{\alpha}\)-spectral radius of bicyclic graphs with given degree sequences. (English) Zbl 1515.05122

Summary: Let \(A(G)\) and \(D(G)\) be the adjacency matrix and the degree matrix of \(G\), respectively. For any real \(\alpha \in [0,1]\), V. Nikiforov [Appl. Anal. Discrete Math. 11, No. 1, 81–107 (2017; Zbl 1499.05384)] defined the matrix \(A_{\alpha}(G)\) as \[ A_{\alpha} (G) = \alpha D(G) + (1-\alpha) A(G). \] In this paper, we generalize some previous results about the \(A_{1/2}\)-spectral radius of bicyclic graphs with a given degree sequence. Furthermore, we characterize all extremal bicyclic graphs which have the largest \(A_{\alpha}\)-spectral radius in the set of all bicyclic graphs with prescribed degree sequences.


05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C12 Distance in graphs
05C07 Vertex degrees


Zbl 1499.05384
Full Text: DOI


[1] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., New York, 1976. · Zbl 1226.05083 · doi:10.1007/978-1-349-03521-2
[2] D. Cvetković, P. Rowlinson and S. Simić, An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts 75, Cambridge University Press, Cambridge, 2010. · Zbl 1211.05002
[3] P. Erdős and T. Gallai, Graphs with prescribed degrees of vertices, Mat. Lapok 11 (1960), 264-274. · Zbl 0103.39701
[4] Y. Huang, B. Liu and Y. Liu, The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences, Discrete Math. 311 (2011), no. 6, 504-511. · Zbl 1222.05130
[5] D. Li, Y. Chen and J. Meng, The \(A_{\alpha} \)-spectral radius of trees and unicyclic graphs with given degree sequence, Appl. Math. Comput. 363 (2019), 124622, 9 pp. · Zbl 1433.05201
[6] D. Li and R. Qin, The \(A_{\alpha} \)-spectral radius of graphs with a prescribed number of edges for \(\frac{1}{2} \leq \alpha \leq 1\), Linear Algebra Appl. 628 (2021), 29-41. · Zbl 1510.05191
[7] H. Lin, X. Huang and J. Xue, A note on the \(A_{\alpha} \)-spectral radius of graphs, Linear Algebra Appl. 557 (2018), 430-437. · Zbl 1396.05073
[8] H. Lin, X. Liu and J. Xue, Graphs determined by their \(A_{\alpha} \)-spectra, Discrete Math. 342 (2019), no. 2, 441-450. · Zbl 1400.05147
[9] H. Lin, J. Xue and J. Shu, On the \(A_{\alpha} \)-spectra of graphs, Linear Algebra Appl. 556 (2018), 210-219. · Zbl 1394.05072
[10] X. Liu and S. Liu, On the \(A_{\alpha} \)-characteristic polynomial of a graph, Linear Algebra Appl. 546 (2018), 274-288. · Zbl 1390.05134
[11] V. Nikiforov, Merging the \(A\)- and \(Q\)-spectral theories, Appl. Anal. Discrete Math. 11 (2017), no. 1, 81-107. · Zbl 1499.05384
[12] V. Nikiforov, G. Pastén, O. Rojo and R. L. Soto, On the \(A_{\alpha} \)-spectra of trees, Linear Algebra Appl. 520 (2017), 286-305. · Zbl 1357.05089
[13] V. Nikiforov and O. Rojo, A note on the positive semidefiniteness of \(A_{\alpha}(G)\), Linear Algebra Appl. 519 (2017), 156-163. · Zbl 1357.05090
[14] ____, On the \(\alpha \)-index of graphs with pendent paths, Linear Algebra Appl. 550 (2018), 87-104. · Zbl 1385.05051
[15] C. Wang and S. Wang, The \(A_{\alpha} \)-spectral radii of graphs with given connectivity, Mathematics 7 (2019), no. 1, 44, 6 pp.
[16] J. Xue, H. Lin, S. Liu and J. Shu, On the \(A_{\alpha} \)-spectral radius of a graph, Linear Algebra Appl. 550 (2018), 105-120. · Zbl 1385.05054
[17] X.-D. Zhang, The Laplacian spectral radii of trees with degree sequences, Discrete Math. 308 (2008), no. 15, 3143-3150. · Zbl 1156.05038
[18] ____, The signless Laplacian spectral radius of graphs with given degree sequences, Discrete Appl. Math. 157 (2009), no. 13, 2928-2937. · Zbl 1213.05153
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.