# Complete Bipartite Graphs

Let’s a have a bipartite complete graph and call it G. Because such graph is bipartite, it has two sets of vertices called respectively set V (G) and set U (G), and the set constituted by its edges is called E(G). Any  edges e is adjacent to a vertex v(V ) and to a vertex u(U ), and, since the graph is bipartite (so it has two subsets of vertices) no vertex is adjacent to any vertex of its same set. Since the graph is complete, every vertex v is adjacent to every vertex u, and every vertex u is adjacent to every vertex v. A vertex is called opinionated if the algebraic difference between the total number of its incident edges labeled 1, and the total number of its incident edges labeled 0, is equal zero. We call a graph G edge-friendly if the algebraic difference between the total number of its incident edges labeled 1 and the total number of its incident edges labeled 0 is equal zero. We show an algorithm that for every positive integers n, m, of the graph Kn,m guarantees to make the graph opinionated and edge-friendly. The label 1 is represented by the color red, and the label 0 is represented by the color black.

While working with bipartite complete graph, our goal is to keep the graph opinionated and edge-friendly. Since the bipartite graphs do not offer clarity in their vision, and most, if they are large, we need to be able to represent them graphically in a more efficient way. We are going to represent them using a matrix. Because the upper vertices of the bipartite graph constitute the set V , and all the down vertices constitute the set U of the graph K( n, n), we need to represent graphically the 2 sets: {v1 ,v2 ,v3 ,vn } and {u1 ,u2 ,u3 ,un }.When all the adjacent edges to the first upper vertex v1 to the down vertices u1 ,u2 ,u3 ,un are reported in the first row .

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