Matrices in the Theory of Signed Simple Graphs (Outline)

This is an expository survey of the uses of matrices in the theory of simple graphs with signed edges. A signed simple graph is a graph, without loops or parallel edges, in which every edge has been declared positive or negative. For many purposes the most significant thing about a signed graph is not the actual edge signs, but the sign of each circle (or ‘cycle’ or ’circuit’), which is the product of the signs of its edges. This fact is manifested in simple operations on the matrices I will present. I treat three kinds of matrices of a signed graph, all of them direct generalizations of familiar matrices from ordinary, unsigned graph theory. The first is the adjacency matrix. The adjacency matrix of an ordinary graph has 1 for adjacent vertices; that of a signed graph has +1 or −1, depending on the sign of the connecting edge. The adjacency matrix leads to questions about eigenvalues and strongly regular signed graphs. The second matrix is the vertex-edge incidence matrix. There are two kinds of incidence matrix of a graph (without signs). The unoriented incidence matrix has two 1’s in each column, corresponding to the endpoints of the edge whose column it is. The oriented incidence matrix has a +1 and a −1 in each column. For a signed graph, there are both kinds of columns, the former corresponding to a negative edge and the latter to a positive edge. Finally, there is the Kirchhoff or Laplacian matrix. This is the adjacency matrix with signs reversed, and with the degrees of the vertices inserted in the diagonal. The Kirchhoff matrix equals the incidence matrix times its transpose. If we multiply in the other order, the transpose times the incidence matrix, we get the adjacency matrix of the line graph, but with 2’s in the diagonal. All this generalizes ordinary graph theory. Indeed, much of graph theory generalizes to signed graphs, while much—though not all—signed graph theory consists of generalizing facts about unsigned graphs. As this is a survey, I will give very few proofs. As it is an outline, I will give few references; they will be added to the final paper.