Abstract
We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.
Metamorphic mechanisms are a class of mechanisms that change their mobility during motions. The deployable and retractable characteristics of these unique mechanisms generate much interest in further investigating their behaviors and potential applications. This paper investigates the resultant configuration variation based on adjacency matrix operation and an improved approach to mechanism synthesis is proposed by adopting elementary matrix operations on the configuration states so as to avoid some omissions caused by existing methods. The synthesis procedure begins with a final configuration of the mechanism, then enumerates possible combinations of different links and associates added links with the binary inverse operations in the matrix transformations of the configuration states, and finally obtains a synthesis result. An algorithm is presented and the topological symmetry of links is used to reduce the number of mechanisms in the synthesis. Some mechanisms of this kind are illustrated as examples.
A note on the eigenvalue free intervals of some classes of signed threshold graphs
