The entries in the main diagnol are a_ii meaning v_i is related to v_i. By related I mean there exists an edge and by v_i I mean the i-th vertex.Suppose that A is a square matrix with each entry equal to 0 or to 1. Prove that A is the adjacency matrix for a dominance digraph if and only if A + the transposition of A has all main diagonal entries equal to 0, and all other entries equal to 1.
Thus, any entry in the main diagnol is a_ii meaning , "is v_i related by v_i"? The answer is no, because that would imply there is an edge from v_i to v_i which is not the case heir because we are not considered looped graphs. Hence everything is zero in the main diagnol. Now, everything else is 1 because because v_i and v_j and i not equal to j we have an edge, as stated in the conditions of the problem. Thus, everything else outside the main diagnol is 1.