On two Laplacian matrices for skew gain graphs - Roshni T. Roy; Shahul Hameed K.; Germina K.A. (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2021)
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.