Constructing a controllable graph under edge constraints

Laplacian Controllability of Interconnected Graphs

“…Similar to the reason in proving Lemma 3.2, the nonzeroness of the first entry of every eigenvector of S suggests that S and S 22 do not share a common eigenvalue. After some row operations, we can write the null space of S − λI as that ofS(λ) defined in (17), whose first row is a zero row. If u TS (λ) = 0 T where u = [ u 1 u 2 · · · u k 1 k 2 ], the row independence of S 22 − λI implies that u (k 1 −1)k 2 +i = 0 for every i ∈ {2, 3, · · · , k 2 }, and thus u (k 1 −1)k 2 +1 = 0.…”

“…In the previous study, we proposed a sufficient condition for a graph to be controllable by one input [17]. The breakthrough of the result is showing that a partial knowledge of the Laplacian spectrum and eigenspaces might be enough to identify a Laplacian controllable graph.…”

“…Specifically, our first result shows that the graph can be generated from the composite graph of a k 1 -vertex graph and a k 2 -vertex graph where k 1 k 2 = k as long as each of these two graphs are Laplacian controllable by an input connected to some vertex of the graph. Our second result provides another solution to generate the graph by interconnecting c antiregular graphs, each with k a vertices, and one k p -vertex path, where ck a + k p = k and c, k a , k p are nonnegative integers (Note that in [17] c is restricted to 1 and the interconnecting scheme is different). As k is large, our results become very helpful in providing a large set of potential solutions for the graph designers to determine the network topology of the system that meets the requirements.…”

Generalising Laplacian controllability of paths

“…The comparisons show that the transition smoothness and the energy consumption in the case of interconnecting two four-vertex antiregular graphs, are between that of a eight-vertex path and that of a eight-vertex antiregular graph (cf. Fig.1 in [19]).…”

Minimal Laplacian controllability problems of threshold graphs

“…This result was later extended to the multi‐input case where more vertices with the same degree were considered [27]. The Laplacian controllability of hybrid graphs constructed by connecting one antiregular graph to another antiregular graph [28] or to a path graph [29] was studied. In this work, we consider the case of generally connected threshold graphs and address their minimal Laplacian controllability issues.…”