The general form of the incidence matrix for the non-normal subgroup graph of generalized quaternion groups

Research of graphs associated to finite groups has been an attractive topic and many new graphs have been defined using some properties of finite groups. One of the graphs is the subgroup graph when the subgroup is not normal. For a group G and a subgroup H which is not normal, the nonnormal subgroup graph is defined as a directed graph with all elements of G are vertex set and two different elements x and y are linked from x to y if their product is in H. There are two well-known matrices representing any graph which are the incidence matrix and the adjacency matrix. Both matrices have different ways of representing the vertices and edges of a graph. In this paper, the general form of the incidence matrix is determined for the nonnormal subgroup of generalized quaternion groups of order 4n. The general form is divided into two cases in which when n is odd and when n is even, respectively. The general form is given in the form of block matrices since the graph is disconnected.