Net Laplacian controllability for joins of signed graphs

Abstract

The net Laplacian matrix of a signed graph Ġ is defined to be NĠ=DĠ±−AĠ, where DĠ± and AĠ are the diagonal matrix of net-degrees and the adjacency matrix of Ġ, respectively. For a binary vector b, the pair (NĠ,b) is controllable if NĠ has no eigenvector orthogonal to b; we also say that Ġ is net Laplacian controllable for b. In this study we consider the net Laplacian controllability of joins of signed graphs. In particular, we establish all controllable pairs (NĠ,b), where Ġ is a signed threshold graph determined by a (0,1,−1)-generating sequence. This result contains all controllable pairs (LG,b), where LG is the Laplacian matrix of a graph G.