On eigenspaces of some compound complex unit gain graphs

Document Type : Workshop on Graphs, Topology and Topological Groups, Cape Town, South Africa


Maurizio Brunetti Dipartimento di Matematica e Applicazioni, Università di Napoli ‘Federico II’, Naples, Italy


Let $\mathbb T$ be the multiplicative group of complex units, and let $L(\Phi)$ denote the Laplacian matrix of a nonempty $\mathbb{T}$-gain graph $\Phi=(\Gamma, \mathbb{T}, \gamma)$. The gain line graph $\mathcal L(\Phi)$ and the gain subdivision graph $\mathcal S(\Phi)$ are defined up to switching equivalence. We discuss how the eigenspaces determined by the adjacency eigenvalues of $\mathcal L(\Phi)$ and $\mathcal S(\Phi)$ are related with those of $L(\Phi)$.


Main Subjects

[1] A. Alazemi, F. Belardo, M. Brunetti, M. Andeli´c and C. M. da Fonseca, Line and subdivision graphs determined
by T4-gain graphs, Mathematics, 7 no. 10 (2019).
[2] F. Belardo and M. Brunetti, Line graphs of complex unit gain graphs with least eigenvalue −2, Electron. J. Linear
Algebra, 37 (2021) 14–30.
[3] F. Belardo, M. Brunetti, M. Cavaleri and A. Donno, Godsil-McKay switching for mixed and gain graphs over the
circle group, Linear Algebra Appl., 614 (2021) 256–269.
[4] F. Belardo, M. Brunetti and A. Ciampella, Signed bicyclic graphs minimizing the least Laplacian eigenvalue, Linear
Algebra Appl., 557 (2018) 201–233.
[5] F. Belardo, M. Brunetti and N. Reff, Balancedness and the least Laplacian eigenvalue of some complex unit gain
graphs, Discuss. Math. Graph Theory, 40 no. 2 (2020) 417–433.
[6] F. Belardo, E. M. Li Marzi and S. K. Simi´c, Signed line graphs with least eigenvalue −2: the star complement
technique, Discrete Appl. Math., 207 (2016) 29–38.
[7] F. Belardo, I. Sciriha and S. K. Simi´c, On eigenspaces of some compound signed graphs, Linear Algebra Appl., 509
(2016) 19–39.
[8] F. Belardo, Z. Stani´c and T. Zaslavsky, Total graph of a signed graph, Math. Contemp., in press (2022), doi:https:
[9] M. Cavaleri, D. D’Angeli and A. Donno, A group representation approach to the balance of gain graph, J. Algebr.
Comb., 54 (2021) 265–293.
[10] M. Cavaleri, D. D’Angeli and A. Donno, Characterizations of line graphs in signed and gain graphs, Eur. J. Comb.,
102 (2022).
[11] M. Cavaleri, D. D’Angeli and A. Donno, Gain-line graphs via G-phases and group representations, Linear Algebra
Appl., 613 (2021) 256–269.
[12] M. Cavaleri and A. Donno, On cospectrality of gain graphs, available at arXiv:2111.12428.
[13] D. Cvetkovi´c, P. Rowlinson and S. Simi´c, Eigenspaces of Graphs, Encyclopedia of Mathematics and its Applications,
66, Cambridge University Press, Cambridge, 1997.
[14] D. Cvetkovi´c, P. Rowlinson and S. K. Simi´c, Graphs with least eigenvalue −2: The star complement technique,
Journal of Algebraic Comb., 14 (2001) 5–16.
[15] D. Cvetkovi´c, P. Rowlinson and S. Simi´c, Spectral Generalizations of Line Graphs, On graphs with least eigenvalue
−2, Cambridge University Press, 2004.
[16] K. Guo and B. Mohar, Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory, 85 no. 1 (2017)
[17] S. He, R.-X. Hao and F. Dong, The rank of a complex unit gain graph in terms of the matching number, Linear
Algebra Appl., 589 (2020) 158–185.
[18] M. Kannan, N. Kumar and S. Pragada, Bounds for the extremal eigenvalues of gain Laplacian matrices, Linear
Algebra Appl., 625 (2021) 212–240.
[19] B. Mohar, A new kind of Hermitian matrices for digraphs, Linear Algebra Appl., 584 (2020) 343–352.
[20] S. Li and W. Wei, The multiplicity of an Aα-eigenvalue: A unified approach for mixed graphs and complex unit
gain graphs, Discrete Math., 343 no. 8 (2020).
[21] L. Lu, J. Wang and Q. Huang, Complex unit gain graphs with exactly one positive eigenvalue, Linear Algebra Appl.,
608 (2021) 270–281.
[22] N. Reff, Spectral properties of complex unit gain graphs, Linear Algebra Appl., 436 no. 9 (2012) 3165–3176.
[23] N. Reff, Oriented gain graphs, line graphs and eigenvalues, Linear Algebra Appl., 506 (2016) 316–328.
[24] I. Sciriha and S. K. Simi´c, On eigenspaces of some compound graphs, in: Recent Results in Designs and Graphs: A
Tribute to Lucia Gionfriddo, Quaderni di Matematica, 28 (2013) 403–417.
[25] Y. Wang, S.-C. Gong and Y.-Z. Fan, On the determinant of the Laplacian matrix of a complex unit gain graph,
Discrete Math., 341 no. 1 (2018) 81–86.
[26] P. Wissing and E. van Dam, Spectral Fundamentals and Characterizations of Signed Directed Graphs, J. Comb.
Theory Ser. A, 187 (2022).
[27] T. Zaslavsky, Biased graphs. I: Bias, balance, and gains, J. Combin. Theory Ser. B, 47 (1989) 32–52.
[28] T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, Electron. J. Combin., Dynamic
Surveys in Combinatorics, 5 (1998) 124 pp.