Controllability of NEPSes of graphs

Abstract

If G is a graph with n vertices, (Formula presented.) is its adjacency matrix and (Formula presented.) is a binary vector of length n, then the pair (Formula presented.) is said to be controllable (or G is said to be controllable for the vector (Formula presented.)) if (Formula presented.) has no eigenvector orthogonal to (Formula presented.). In particular, if (Formula presented.) is the all-1 vector (Formula presented.), then we simply say that G is controllable. In this paper, we consider the controllability of non-complete extended p-sums (for short, NEPSes) of graphs. We establish some general results and then focus the attention to the controllability of paths and related NEPSes. Moreover, the controllability of Cartesian products and tensor products is also considered. Certain related results concerning signless Laplacian matrices and signed graphs are reported.