Title: Matrices for which zero and nonzero suffice
Abstract: Graphs illustrate how a group of objects are interconnected. Recently, graphs have been used to study computer, neural, and social networks. Matrices, in particular their eigenvalues, play a major role in the analysis of graphs. For example the eigenvalues of the adjacency matrix can tell you how many triangles are in the graph or the “energy” of a graph. In some cases the multiplicity of an eigenvalue is more important than the eigenvalue itself. This is true for generalized Laplacian matrices and the role they play in determining the planarity of its corresponding graph.
The adjacency matrix is a 0,1-matrix in which each 1 represents an edge while each 0 signals the absence of an edge. Generalizing this idea, each graph G has a family of symmetric matrices, S(G), which are naturally associated with it. Off-diagonal entries of matrices in S(G) are nonzero if and only if there is a corresponding edge in G. The eigenvalues of matrices in S(G) are of interest and in particular: What is the maximum multiplicity of an eigenvalue for matrices in S(G)? I will share some results concerning this question and some of its variants.