Complete Bipartite Graphs

On Opinionated Complete Bipartite Graphs
Laura Parrish
[email protected]
Joint work with
and
June 18, 2013
Abstract
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 black1 Introduction
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 of…