### On two Laplacian matrices for skew gain graphs

#### Abstract

Gain graphs are graphs where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. Generalizing the notion of gain graphs, skew gain graphs have the property that the gain of a reversed edge is the image of edge gain under an anti-involution. In this paper, we study two different types, Laplacian and *g*-Laplacian matrices for a skew gain graph where the skew gains are taken from the multiplicative group *F ^{x}* of a field

*F*of characteristic zero. Defining incidence matrix, we also prove the matrix tree theorem for skew gain graphs in the case of the

*g*-Laplacian matrix.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2021.9.1.12

#### References

N. Biggs, Algebraic graph theory, Cambrige University Press, Cambridge (1974).

J.G. Broida and S.G. Williamson, Comprehensive introduction to linear algebra, Addison Wesley, Redwood City (1989).

S. Chaiken, A combinatorial proof of the all minors matrix tree theorem, SIAM J. Algebraic Discrete Methods 3 (1982), 319–329.

C. Delorme, Weighted graphs: Eigenvalues and chromatic number, Electron. J. Graph Theory Appl. 4 (1) (2016), 8–17.

H.A. Ganie, S. Pirzada, and E.T. Baskoro, On energy, Laplacian energy and p-fold graphs, Electron. J. Graph Theory Appl. 3 (1) (2015), 94–107.

J. Hage, The membership problem for switching classes with skew gains, Fundamenta Informaticae 39 (4) (1999), 375–387.

J. Hage and T. Harju, The size of switching classes with skew gains, Discrete Math. 215 (2000), 81–92.

F. Harary, Graph theory, Addison Wesley, Reading Massachusetts (1969).

R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra and its Applications 197-198 (1994), 143–176.

K. Shahul Hameed and K.A. Germina, Balance in gain graphs-A spectral analysis, Linear Algebra and its Applications 436 (2012), 1114–1121.

K. Shahul Hameed, R.T. Roy, P. Soorya, and K.A. Germina, On the characteristic polynomial of skew gain graphs, communicated (2020).

T. Zaslavsky, Biased graphs. I. Bias, balance, and gains, J. Combin. Theory Ser. B 47 (1989), 32–52.

T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), 47–74.

### Refbacks

- There are currently no refbacks.

ISSN: 2338-2287

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.