Balance theory: An extension to conjugate skew gain graphs

Document Type : Original paper

Authors

Department of Mathematics K M M Government Women's College Kannur, Kerala 670 004 India

Abstract

We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the $g$-Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs.

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References

[1] J. Hage and T. Harju, The size of switching classes with skew gains, Discrete Math. 215 (2000), no. 1-3, 81–92.
https://doi.org/10.1016/S0012-365X(99)00243-5
[2] F. Harary, The determinant of the adjacency matrix of a graph, Siam Review 4 (1962), no. 3, 202–210.
https://doi.org/10.1137/1004057
[3] F. Harary, Graph Theoryn, Addison Wesley, Reading, Mass., 1972.
[4] S.H. Koombail and K.A. Germina, Balance in gain graphs–A spectral analysis, Linear algebra Appl. 436 (2012), no. 5, 1114–1121.
https://doi.org/10.1016/j.laa.2011.07.005
[5] S.H. Koombail and K.A. Germina, On the characterisitic polynomial of skew gain graphs, Southeast Asian Bull. Math. (to appear).
[6] R. Mehatari, M.R. Kannan, and A. Samanta, On the adjacency matrix of a complex unit gain graph, Linear Multilinear Algebra 70 (2022), no. 9, 1798–1813.
https://doi.org/10.1080/03081087.2020.1776672
[7] N. Reff, Spectral properties of complex unit gain graphs, Linear Algebra Appl. 436 (2012), no. 9, 3165–3176.
https://doi.org/10.1016/j.laa.2011.10.021.
[8] R.T. Roy, S.H. Koombail, and K.A. Germina, On two Laplacian matrices for skew gain graphs, Electron. J. Graph Theory Appl. 9 (2021), no. 1, 125–135.
https://dx.doi.org/10.5614/ejgta.2021.9.1.12
[9] T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), no. 1, 47–74. Erratum. Discrete Appl. Math. 5 (1983) 47–74.
https://doi.org/10.1016/0166-218X(82)90033-6
[10] T. Zaslavsky, Biased graphs. I. Bias, balance, and gains, J. Combin. Theory, Ser. B 47 (1989), no. 1, 32–52.
http://doi.org/10.1016/0095-8956(89)90063-4
 
Communications in Combinatorics and Optimization
Volume 9, Issue 2
June 2024
Pages 253-262

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