The incidence matrix and labelings of a graph

AbstractWe deal with the problem of labeling the edges of a graph in such a way that the labels of the edges incident with any vertex add up to a value prescribed for that vertex. We show that the use of elementary column operations on the incidence matrix is fruitful in giving easy proofs of theorems on magic graphs and labeling |1, 3, 4|. The method can be visualized in the graph and also leads to a simple proof of a theorem in |2| on the multiplicity of −2 as an eigenvalue of a line graph. We also deal with mixed graphs, the label of a directed edge being subtracted at its initial vertex