Let $ G $ be a directed graph with $ \mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\; n. $ The adjacency matrix of the directed graph $ G $ is a matrix $ A = \left[{a}_{ij}\right] $ of order $ n\times n, $ such that for $ i\ne j $, if there is an arc from $ i $ to $ j, $ then $ {a}_{ij} = 1 $, otherwise $ {a}_{ij} = 0 $. Matrix $ B = J-A $ is called the antiadjacency matrix of the directed graph $ G, $ where $ J $ is the matrix of order $ n\times n $ with all of those entries are one. In this paper, we provided several properties of the adjacency matrices of directed graphs, such as a determinant of a directed graphs, the characteristic polynomial of acyclic directed graphs, and regular directed graphs. Moreover, we discuss antiadjacency energy of acyclic directed graphs and give some examples of antiadjacency energy for several families of graphs.
Citation: Kiki A. Sugeng, Fery Firmansah, Wildan, Bevina D. Handari, Nora Hariadi, Muhammad Imran. Several properties of antiadjacency matrices of directed graphs[J]. AIMS Mathematics, 2024, 9(10): 27834-27847. doi: 10.3934/math.20241351
Let $ G $ be a directed graph with $ \mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\; n. $ The adjacency matrix of the directed graph $ G $ is a matrix $ A = \left[{a}_{ij}\right] $ of order $ n\times n, $ such that for $ i\ne j $, if there is an arc from $ i $ to $ j, $ then $ {a}_{ij} = 1 $, otherwise $ {a}_{ij} = 0 $. Matrix $ B = J-A $ is called the antiadjacency matrix of the directed graph $ G, $ where $ J $ is the matrix of order $ n\times n $ with all of those entries are one. In this paper, we provided several properties of the adjacency matrices of directed graphs, such as a determinant of a directed graphs, the characteristic polynomial of acyclic directed graphs, and regular directed graphs. Moreover, we discuss antiadjacency energy of acyclic directed graphs and give some examples of antiadjacency energy for several families of graphs.
| [1] | R. Bapat, Graphs and matrices, London: Springer, New Delhi: Hindustan Book Agency, 2014. https://doi.org/10.1007/978-1-4471-6569-9 |
| [2] |
J. Wang, M. Lu, F. Belardo, M. Randić, The anti-adjacency matrix of a graph: eccentricity matrix, Discrete Appl. Math. , 251 (2018), 299–309. https://doi.org/10.1016/j.dam.2018.05.062 doi: 10.1016/j.dam.2018.05.062
|
| [3] | G. Chartrand, L. Lesniak, P. Zhang, Graphs & digraphs, 6 Eds., New York: Chapman and Hall/CRC, 2015. https://doi.org/10.1201/b19731 |
| [4] | A. Brouwer, W. Haemers, Spectra of graphs, New York: Springer, 2011. https://doi.org/10.1007/978-1-4614-1939-6 |
| [5] | C. Meyer, Matrix analysis and applied linear algebra, 2 Eds., New Jersey: SIAM, 2000. https://doi.org/10.1137/1.9781611977448 |
| [6] |
H. Ramane, H. Walkar, S. Rao, B. Acharya, P. Hampiholi, S. Jog, et al., Spectra and energies of iterated line graphs of regular graphs, Appl. Math. Lett. , 18 (2005), 679–682. https://doi.org/10.1016/j.aml.2004.04.012 doi: 10.1016/j.aml.2004.04.012
|
| [7] |
D. Diwyacitta, A. Putra, K. Sugeng, S. Utama, The determinant of an antiadjacency matrix of a directed cycle graph with chords, AIP Conf. Proc. , 1862 (2017), 030127. https://doi.org/10.1063/1.4991231 doi: 10.1063/1.4991231
|
| [8] |
R. Stanley, A matrix for counting paths in acyclic digraphs, J. Comb. Theory A, 74 (1996), 169–172. https://doi.org/10.1006/jcta.1996.0046 doi: 10.1006/jcta.1996.0046
|
| [9] |
M. Edwina, K. Sugeng, Determinant of anti-adjacency matrix of union and join operation from two disjoint of several classes of graphs, AIP Conf. Proc. , 1862 (2017), 030158. https://doi.org/10.1063/1.4991262 doi: 10.1063/1.4991262
|
| [10] |
B. Aji, K. Sugeng, S. Aminah, Characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic corona graph, J. Phys.: Conf. Ser. , 1836 (2021), 012001. https://doi.org/10.1088/1742-6596/1722/1/012055 doi: 10.1088/1742-6596/1722/1/012055
|
| [11] |
M. Prayitno, S. Utama, S. Aminah, Properties of anti-adjacency matrix of directed cyclic sun graph, IOP Conf. Ser.: Mater. Sci. Eng., 567 (2019), 012020. https://doi.org/10.1088/1757-899X/567/1/012020 doi: 10.1088/1757-899X/567/1/012020
|
| [12] | M. Solihin, S. Aminah, S. Utama, Properties of anti-adjacency matrix of cyclic directed windmill graph $ K^{\xrightarrow{ }}(4, N) $, Proceedings of the Mathematics, Informatics, Science, and Education International Conference, 2018, 9–12. https://doi.org/10.2991/Miseic-18.2018.3 |
| [13] | G. Putra, Characteristic polynomial and eigenvalues of antiadjacency matrix for graph $ {K}_{m}\odot{K}_{1} $ and $ {H}_{m}\odot{K}_{1} $, Sitekin: Jurnal Sains, Teknologi dan Industri, 21 (2024), 357–361. |
| [14] |
W. Irawan, K. Sugeng, Characteristic antiadjacency matrix of graph join, BAREKENG: J. Math. Appl. , 16 (2022), 041–046. https://doi.org/10.30598/barekengvol16iss1pp041-046 doi: 10.30598/barekengvol16iss1pp041-046
|
| [15] |
J. Daniel, K. Sugeng, N. Hariadi, Eigenvalues of antiadjacency matrix of cayley graph of $ {\mathrm{Z}}_{\mathrm{n}} $, Indonesian Journal of Combinatorics, 6 (2022), 66–76. https://doi.org/10.19184/ijc.2022.6.1.5 doi: 10.19184/ijc.2022.6.1.5
|
| [16] |
R. Balakrishnan, The energy of a graph, Linear Algebra Appl. , 387 (2004), 287–295. https://doi.org/10.1016/j.laa.2004.02.038 doi: 10.1016/j.laa.2004.02.038
|
Figures(1) / Tables(4)
Kiki A. Sugeng, Fery Firmansah, Wildan, Bevina D. Handari, Nora Hariadi, Muhammad Imran. Several properties of antiadjacency matrices of directed graphs[J]. AIMS Mathematics, 2024, 9(10): 27834-27847. doi: 10.3934/math.20241351
DownLoad: