On two Laplacian matrices for skew gain graphs

Roshni T. Roy (Department of Mathematics, Central University of Kerala, India)
Shahul Hameed K. (Department of Mathematics, K M M Government Women’s College, Kannur - 670004, Kerala, India)
Germina K.A. (Department of Mathematics, Central University of Kerala, India)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 9, No 1 (2021): Electronic Journal of Graph Theory and Applications

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 Fx 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.