Spectral Properties of Dual Unit Gain Graphs
Abstract
:1. Introduction
2. Gain Graphs, Dual Elements and Dual Matrices
2.1. Gain Graphs
- (i)
- Φ is balanced.
- (ii)
- .
- (iii)
- φ has a potential function.
2.2. Dual Elements
- (i)
- and the equality holds if, and only if, is a nonnegative dual number.
- (ii)
- and .
- (iii)
- .
- (iv)
- .
- (v)
- .
- (i)
- For any unit dual complex number , there is a dual angle such that . Here, .
- (ii)
- For any positive integer n, there is for any
- (iii)
2.3. Dual Matrices
- (i)
- For any vector , there exists such that .
- (ii)
- Suppose that and , where and . Then, .
3. Dual Unit Gain Graphs
- (i)
- Φ is balanced.
- (ii)
- .
- (iii)
- φ has a potential function.
- (i)
- There exists a switching function such that for and .
- (ii)
- (iii)
- (i)
- Define the switching function as follows:Then, there isFor , there isFor , there is
- (ii)
- The proof is a slight modification of Theorem 6.1 in [13]. LetFurthermore, we have and . Thus, the eigenvalue of A is equal to . By Theorem 2, there is . This derives the closed form of . The closed form of follows directly from .
- (iii)
- By Theorem 1, there exists such that . Let for . Then, there is such that for and . The result of this item follows directly from that of item (ii).
4. Eigenvalues of Adjacency Matrices
- (i)
- The eigenvalue satisfies
- (ii)
- is an eigenvalue of the matrix with an eigenvector .
- (iii)
- The dual part satisfies . Furthermore, if has n simple eigenvalues with associated unit eigenvectors ’s. Then, A has exactly n eigenvalues with associated eigenvectors , , where
- (iv)
- Two eigenvectors of A, associated with two eigenvalues with distinct standard parts, are orthogonal to each other.
5. Coefficient Theorems of Characteristic Polynomials
6. Eigenvalues of Laplacian Matrices
7. Numerical Experiments
7.1. Three-Points Cycles
7.2. Large-Scale Examples
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Cavaleri, M.; D’Angeli, D.; Donno, A. A group representation approach to balance of gain graphs. J. Algebr. Comb. 2021, 54, 265–293. [Google Scholar] [CrossRef]
- Shahul Hameed, K.; Germina, K.A. Balance in gain graphs—A spectral analysis. Linear Algebra Its Appl. 2012, 436, 1114–1121. [Google Scholar] [CrossRef]
- Reff, N. Oriented gain graphs, line graphs and eigenvalues. Linear Algebra Its Appl. 2016, 506, 316–328. [Google Scholar] [CrossRef]
- Harary, F. On the notion of balanced in a signed graph. Mich. Math. J. 1953, 2, 143–146. [Google Scholar] [CrossRef]
- Zaslavsky, T. Signed graphs. Discret. Appl. Math. 1982, 4, 47–74. [Google Scholar] [CrossRef]
- Zaslavsky, T. Biased graphs. I. Bias, balance, and gains. J. Comb. Theory Ser. B 1989, 47, 32–52. [Google Scholar] [CrossRef]
- Hou, Y.; Li, J.; Pan, Y. On the Laplacian eigenvalues of signed graphs. Linear Multilinear Algebra 2003, 51, 21–30. [Google Scholar] [CrossRef]
- Akbari, S.; Belardo, F.; Heydari, F.; Maghasedi, M.; Souri, M. On the largest eigenvalue of signed unicyclic graphs. Linear Algebra Its Appl. 2019, 581, 145–162. [Google Scholar] [CrossRef]
- Fan, Y. On the least eigenvalue of a unicyclic mixed graph. Linear Multilinear Algebra 2005, 53, 97–113. [Google Scholar] [CrossRef]
- Hou, Y. Bounds for the least Laplacian eigenvalue of a signed graph. Acta Math. Sin. Engl. Ser. 2005, 21, 955–960. [Google Scholar] [CrossRef]
- Belardo, F. Balancedness and the least eigenvalue of Laplacian of signed graphs. Linear Algebra Its Appl. 2014, 446, 133–147. [Google Scholar] [CrossRef]
- Yu, G.; Yuan, X.; Qu, H. Signed k-uniform hypergraphs and tensors. Linear Algebr. Its Appl. 2019, 580, 1–13. [Google Scholar] [CrossRef]
- Reff, N. Spectral properties of complex unit gain graphs. Linear Algebra Its Appl. 2012, 436, 3165–3176. [Google Scholar] [CrossRef]
- Bapat, R.B.; Kalita, D.; Pati, S. On weighted directed graphs. Linear Algebra Its Appl. 2012, 436, 99–111. [Google Scholar] [CrossRef]
- Alazemi, A.; Andelić, M.; Belardo, F.; Brunetti, M.; da Fonseca, C.M. Line and subdivision graphs determined by -gain graphs. Mathematics 2019, 7, 926. [Google Scholar] [CrossRef]
- He, S.; Hao, R.; Dong, F. The rank of a complex unit gain graph in terms of the matching number. Linear Algebra Its Appl. 2020, 589, 158–185. [Google Scholar] [CrossRef]
- Li, S.; Wei, W. The multiplicity of an Aα-eigenvalue: A unified approach for mixed graphs and complex unit gain graphs. Discret. Math. 2020, 343, 111916. [Google Scholar] [CrossRef]
- Wang, Y.; Gong, S.C.; Fan, Y.Z. On the determinant of the Laplacian matrix of a complex unit gain graph. Discret. Math. 2018, 341, 81–86. [Google Scholar] [CrossRef]
- Belardo, F.; Brunetti, M.; Coble, N.J.; Reff, N.; Skogman, H. Spectra of quaternion unit gain graphs. Linear Algebra Its Appl. 2022, 632, 15–49. [Google Scholar] [CrossRef]
- Kyrchei, I.I.; Treister, E.; Pelykh, V.O. The determinant of the Laplacian matrix of a quaternion unit gain graph. Discret. Math. 2024, 147, 113955. [Google Scholar] [CrossRef]
- Zhou, Q.; Lu, Y. Relation between the row left rank of a quaternion unit gain graph and the rank of its underlying graph. Electron. J. Linear Algebra 2023, 39, 181–198. [Google Scholar] [CrossRef]
- Clifford, W.K. Preliminary sketch of bi-quaternions. Proc. Lond. Math. Soc. 1873, 4, 381–395. [Google Scholar]
- Study, E. Geometrie der Dynamen; Verlag Teubner: Leipzig, Germany, 1903. [Google Scholar]
- Abaunza, H.; Castillo, P.; Victorino, A.; Lozano, R. Dual quaternion modeling and control of a quad-rotor aerial manipulator. J. Intell. Robot. Syst. 2017, 88, 267–283. [Google Scholar] [CrossRef]
- Chen, Z.; Ling, C.; Qi, L.; Yan, H. A regularization-patching dual quaternion optimization method for solving the hand-eye calibration problem. J. Optim. Theory Appl. 2024, 200, 1193–1215. [Google Scholar] [CrossRef]
- Daniilidis, K. Hand-eye calibration using dual quaternions. Int. J. Robot. Res. 1999, 18, 286–298. [Google Scholar] [CrossRef]
- Wang, X.; Yu, C.; Lin, Z. A dual quaternion solution to attitude and position control for rigid body coordination. IEEE Trans. Robot. 2012, 28, 1162–1170. [Google Scholar] [CrossRef]
- Wei, T.; Ding, W.; Wei, Y. Singular value decomposition of dual matrices and its application to traveling wave identification in the brain. SIAM J. Matrix Anal. Appl. 2024, 45, 634–660. [Google Scholar] [CrossRef]
- Cheng, J.; Kim, J.; Jiang, Z.; Che, W. Dual quaternion-based graph SLAM. Robot. Auton. Syst. 2016, 77, 15–24. [Google Scholar] [CrossRef]
- Qi, L.; Cui, C. Eigenvalues of dual Hermitian matrices with application in formation control. arXiv 2024, arXiv:2403.10308v2. To appear in SIAM J. Matrix Anal. Appl. [Google Scholar]
- Qi, L.; Ling, C.; Yan, H. Dual quaternions and dual quaternion vectors. Commun. Appl. Math. Comput. 2022, 4, 1494–1508. [Google Scholar] [CrossRef]
- Zhang, F. Quaternions and matrices of quaternions. Linear Algebra Its Appl. 1997, 251, 21–57. [Google Scholar] [CrossRef]
- Pennestrì, E.; Stefanelli, R. Linear algebra and numerical algorithms using dual numbers. Multibody Syst. Dyn. 2007, 18, 323–344. [Google Scholar] [CrossRef]
- Qi, L.; Cui, C. Eigenvalues and Jordan forms of dual complex matrices. Commun. Appl. Math. Comput. 2023, 1–17. [Google Scholar] [CrossRef]
- Qi, L.; Luo, Z. Eigenvalues and singular values of dual quaternion matrices. Pac. J. Optim. 2023, 19, 257–272. [Google Scholar]
- Chen, Y.; Zhang, L. Generalized Jacobi method for computing eigenvalues of dual quaternion Hermitian matrices. arXiv 2024, arXiv:2405.13649v3. [Google Scholar]
- Cui, C.; Qi, L. A power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. J. Sci. Comput. 2024, 100, 21. [Google Scholar] [CrossRef]
- Ding, W.; Li, Y.; Wang, T.; Wei, M. Dual quaternion singular value decomposition based on bidiagonalization to a dual number matrix using dual quaternion Householder transformations. Appl. Math. Lett. 2024, 152, 109021. [Google Scholar] [CrossRef]
- Duan, A.Q.; Wang, Q.W.; Duan, X.F. On Rayleigh quotient iteration for dual quaternion Hermitian eigenvalue problem. arXiv 2024, arXiv:2310.20290v6. [Google Scholar]
- Ling, C.; Qi, L.; Yan, H. Minimax principle for eigenvalues of dual quaternion Hermitian matrices and generalized inverses of dual quaternion matrices. Numer. Funct. Anal. Optim. 2023, 44, 1371–1394. [Google Scholar] [CrossRef]
- Stanić, Z. Integral regular net-balanced signed graphs with vertex degree at most four. Ars Math. Contemp. 2019, 17, 103–114. [Google Scholar] [CrossRef]
- Mehatari, R.; Kannan, M.R.; Samanta, A. On the adjacency matrix of a complex unit gain graph. Linear Multilinear Algebra 2022, 70, 1798–1813. [Google Scholar] [CrossRef]
- Qi, L.; Wang, X.; Luo, Z. Dual quaternion matrices in multi-agent formation control. Commun. Math. Sci. 2023, 21, 1865–1874. [Google Scholar] [CrossRef]
- Moore, E.H. On the determinant of an hermitian matrix of quaternionic elements. Bull. Am. Math. Soc. 1922, 28, 161–162. [Google Scholar]
- Kyrchei, I.I. The theory of the column and row determinants in a quaternion linear algebra. In Advances in Mathematical Research; Baswell, A.R., Ed.; Nova Science Publishers: New York, NY, USA, 2012; Volume 15, pp. 301–358. [Google Scholar]
- Aslaksen, H. Quaternionic determinants. Math. Intell. 1996, 18, 57–65. [Google Scholar] [CrossRef]
- Cui, C.; Qi, L.; Song, G.; Wang, Q. Moore determinant of dual quaternion Hermitian matrices. Comput. Appl. Math. 2024, 43, 365. [Google Scholar] [CrossRef]
- Cvetković, D.M.; Doob, M.; Sachs, H. Spectra of Graphs: Theory and Applications, 3rd ed.; Johann Ambrosius Barth: Heidelberg, Germany, 1995. [Google Scholar]
- Belardo, F.; Cioabǎ, S.M.; Koolen, J.; Wang, J. Open problems in the spectral theory of signed graphs. Art Discret. Appl. Math. 2018, 1, P2.10. [Google Scholar] [CrossRef]
DCUGG | |||||||
---|---|---|---|---|---|---|---|
Method | 10 | 20 | 50 | 100 | 200 | 500 | |
(10) | CPU (s) | 7.06 | 2.50 | 6.20 | 1.38 | 1.70 | 9.81 |
SMM | CPU (s) | 5.67 | 1.13 | 1.20 | 9.93 | 7.47 | 1.49 |
RES | 1.77 | 2.80 | 1.83 | 1.62 | 1.00 | 6.03 | |
DQUGG | |||||||
(10) | CPU (s) | 5.44 | 3.03 | 6.26 | 1.13 | 2.32 | 5.32 |
SMM | CPU (s) | 9.10 | 2.25 | 8.93 | 3.15 | 2.10 | 4.86 |
RES | 3.71 | 2.68 | 1.13 | 1.59 | 6.89 | 3.00 |
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Cui, C.; Lu, Y.; Qi, L.; Wang, L. Spectral Properties of Dual Unit Gain Graphs. Symmetry 2024, 16, 1142. https://doi.org/10.3390/sym16091142
Cui C, Lu Y, Qi L, Wang L. Spectral Properties of Dual Unit Gain Graphs. Symmetry. 2024; 16(9):1142. https://doi.org/10.3390/sym16091142
Chicago/Turabian StyleCui, Chunfeng, Yong Lu, Liqun Qi, and Ligong Wang. 2024. "Spectral Properties of Dual Unit Gain Graphs" Symmetry 16, no. 9: 1142. https://doi.org/10.3390/sym16091142