Abstract
Let G be a graph of order n with adjacency matrix A(G) and diagonal matrix of degree D(G). For every \(\alpha \in [0,1]\), Nikiforov (Appl Anal Discrete Math 11(1):81–107, 2017) defined the matrix \(A_\alpha (G) = \alpha D(G) + (1-\alpha )A(G)\). In this paper, we present the \(A_{\alpha }(G)\)-characteristic polynomial when G is obtained by coalescing two graphs, and if G is a semi-regular bipartite graph we obtain the \(A_{\alpha }\)-characteristic polynomial of the line graph associated with G. Moreover, if G is a regular graph, we exhibit the \(A_{\alpha }\)-characteristic polynomial for the graphs obtained from some operations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brondani AE, Oliveira CS, França FAM, de Lima L (2019) A\(_\alpha \)-spectrum of a firefly graph. Electron Notes Theor Comput Sci 346:209–219
Brondani AE, França FAM, Oliveira CS (2022) Positive semidefiniteness of A\(_\alpha \)(G) on some families of graphs. Discrete Appl Math 323:113–123
Chen Y, Li D, Meng J (2019) On the second largest A\(_\alpha \)(G)-eigenvalues of graphs. Linear Algebra Appl 580:343–358
Cvetković D, Rowlinson P, Simić S (2010) London Mathematical Society student texts. An introduction to the theory of graph spectra. Cambridge University Press, Cambridge
Francis A, Smith D, Webb B (2019) General equitable decompositions for graphs with symmetries. Linear Algebra Appl 577:287–316. https://doi.org/10.1016/j.laa.2019.04.035
Goldberg K (1959) Principal sub-matrices of a full-rowed non-negative matrix. J Res Natl Bureau Stand Sect B Math Math Phys 63B(1):19–20
Horn RA, Johnson CR (2013) Matrix analysis. Cambridge University Press, New York
Li S, Wang S (2019) The A\(_\alpha \)- spectrum of graph product. Electron J Linear Algebra 35:473–481
Lin H, Liu X, Xue J (2017) Graphs determined by their A\(_\alpha \)-spectra. arXiv: Combinatorics
Lin H, Huang X, Xue J (2018) A note on the A\(_\alpha \)-spectral radius of graphs. Linear Algebra Appl 557:430–437
Lin H, Xue J, Shu J (2018) On the A\(_\alpha \)-spectra of graphs. Linear Algebra Appl 556:210–219
Liu X, Liu S (2018) On the A\(_\alpha \)-characteristic polynomial of a graph. Linear Algebra Appl 546:274–288
Liu S, Das KC, Shu J (2020) On the eigenvalues of A\(_\alpha \)-matrix of graphs. Discrete Math 343(8):111917
Liu S, Das KC, Sun S, Shu J (2020) On the least eigenvalue of A\(_\alpha \)-matrix of graphs. Linear Algebra Appl 586:347–376
Nikiforov V (2017) Merging the A- and Q- spectral theories. Appl Anal Discrete Math 11(1):81–107
Nikiforov V, Rojo O (2017) A note on the positive semidefiniteness of A\(_\alpha \). Linear Algebra Appl 519:156–163
Pirzada S (2022) Two upper bounds on the A\(_\alpha \)-spectral radius of a connected graph. Commun Combin Optim 7(1):53–57
Silvester JR (2000) Determinants of block matrices. Math Gazette 84(501):460–467
Tahir MA, Zhang X-D (2018) Graphs with three distinct \(\alpha \)-eigenvalues. Acta Math Vietnam 43(4):649–659
Tahir MA, Zhang X-D (2020) Coronae graphs and their \(\alpha \)-eigenvalues. Bull Malays Math Sci Soc 43:2911–2927
Wang S, Wong D, Tian F (2020) Bounds for the largest and the smallest A\(_\alpha \) eigenvalues of a graph in terms of vertex degrees. Linear Algebra Appl 590:210–223
Acknowledgements
The research of C. S. Oliveira is supported by CNPq Grant 304548/2020-0.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Leonardo de Lima.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
da Silva Junior, J.D.G., Oliveira, C.S. & da Costa, L.M.G.C. On the characteristic polynomial of the \(A_\alpha \)-matrix for some operations of graphs. Comp. Appl. Math. 42, 206 (2023). https://doi.org/10.1007/s40314-023-02329-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02329-3