Characteristic Polynomial and Eigenvalues of Anti-adjacency Matrix for Graph K_m ⨀ K_1 and H_m ⨀ K_1
DOI:
https://doi.org/10.24014/sitekin.v21i2.30899Abstract
Let G=(V,E) be a simple and connected graph. The adjacency matrix G is a representation of a graph in the form of a square matrix, with the size of the matrix determined by the order G. By defining a graph into a matrix, lots of research related to a spectrum has been done by researchers. Later, they defined the anti-adjacency matrix as a matrix obtained by subtracting a matrix with all entries equal to one and the adjacency matrix G. In this paper, we determine the characteristic polynomial of matrix anti-adjacency for corona product between complete graph K_m and K_1and hyper-octahedral graph H_m and K_1 with the eigenvalues.
References
R. Diestel, Graph Theory, Fifth. Germany: Springer-Verlag, 2017.
N. Biggs, Algebraic Graph Theory, Second. London: Cambridge University Press, 1993. doi: https://doi.org/10.1017/CBO9780511608704.
W. Irawan and K. A. Sugeng, “Characteristic Antiadjacency Matrix of Graph Join,” BAREKENG J. Ilmu Mat. dan Terap., vol. 16, no. 1, pp. 041–046, 2022, doi: 10.30598/barekengvol16iss1pp041-046.
R. Frucht and F. Harary, “On the corona of two graphs,” an equation Math., vol. 4, no. 3, pp. 322–325, 1970, doi: 10.1007/BF01844162.
D. Diwyacitta, A. P. Putra, K. A. Sugeng, and S. Utama, “The determinant of an antiadjacency matrix of a directed cycle graph with chords,” AIP Conf. Proc., vol. 1862, 2017, doi: 10.1063/1.4991231.
M. Edwina and K. A. Sugeng, “Determinant of the anti adjacency matrix of union and join operation from two disjoint of several classes of graphs,” AIP Conf. Proc., vol. 1862, 2017, doi: 10.1063/1.4991262.
L. Widiastuti, S. Utama, and S. Aminah, “Characteristic Polynomial of Antiadjacency Matrix of Directed Cyclic Wheel Graph $overrightarrow {{W_n}},” J. Phys. Conf. Ser., vol. 1108, p. 12009, Nov. 2018, doi: 10.1088/1742-6596/1108/1/012009.
H. B. Aji, K. Sugeng, and S. Aminah, “Characteristic polynomial and eigenvalues of anti adjacency matrix of directed unicyclic flower vase graph,” J. Phys. Conf. Ser., vol. 1722, p. 12055, Jan. 2021, doi: 10.1088/1742-6596/1722/1/012055.
N. Anzana, S. Aminah, and S. Utama, “Characteristic polynomial of anti-Adjacency matrix of directed cyclic friendship graph,” J. Phys. Conf. Ser., vol. 1538, no. 1, pp. 1–7, 2020, doi: 10.1088/1742-6596/1538/1/012007.
N. Hasyyati, K. A. Sugeng, and S. Aminah, “Characteristic polynomial and eigenvalues of anti-adjacency matrix of directed unicyclic corona graph,” J. Phys. Conf. Ser., vol. 1836, no. 1, p. 12001, 2021, doi: 10.1088/1742-6596/1836/1/012001.
M. Prayitno and K. Sugeng, “On Characteristic Polynomial of Antiadjacency Matrix of A Line Digraph,” J. Mat. UNAND, vol. 11, p. 74, Apr. 2022, doi: 10.25077/jmu.11.1.74-81.2022.
F. Zhang, Matrix Theory: Basic Results and Techniques, Second. London: Springer, 2011. doi: 10.1007/978-1-4614-1099-7.
S. H. Friedberg, A. J. Insel, and Lawrence E. Spence, Linear Algebra, Fifth. London: Pearson, 2018.
H. Anton and C. Rorres, Elementary Linear Algebra: Applications Version, 11th Edition. John Wiley & Sons Incorporated, 2013. [Online]. Available: https://books.google.co.id/books?id=loRbAgAAQBAJ
C. Godsil and G. Royle, “Graphs BT - Algebraic Graph Theory,” C. Godsil and G. Royle, Eds. New York, NY: Springer New York, 2001, pp. 1–18. doi: 10.1007/978-1-4613-0163-9_1.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License
Copyright Notice
An author who publishes in the SITEKIN Journal agrees to the following terms:
- Author retains the copyright and grants the journal the right of first publication of the work simultaneously licensed under the Creative Commons Attribution-ShareAlike 4.0 License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal
- Author is able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book) with the acknowledgement of its initial publication in this journal.
- Author is permitted and encouraged to post his/her work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of the published work (See The Effect of Open Access).
Read more about the Creative Commons Attribution-ShareAlike 4.0 Licence here: https://creativecommons.org/licenses/by-sa/4.0/.
