This paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory. The vehicles exchange information according to a pre-specified communication digraph, G. A feedback control is designed using relative information between a vehicle and its in-neighbors in G. We prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that G has a rooted directed spanning tree. We show the direct relationship between the rate of convergence to formation and the eigenvalues of the (directed) Laplacian of G. Various special situations are discussed, including symmetric communication graphs and formations with leaders. Several numerical simulations are used to illustrate the results.

Let $G$ denote a directed graph with adjacency matrix $Q$ and in-degree matrix $D$. We consider the Kirchhoff matrix $L=D-Q$, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when $G$ is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of $G$. This fact has a meaningful generalization to directed graphs, as was recently observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace – namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights.

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Let denote a distance-regular graph with vertex set X , diameter D ≥ 3, valency k ≥ 3, and assume supports a spin modelTo avoid degenerate situations we assume is not a Hamming graph and t i ∈ {t 0 , −t 0 } for 1 ≤ i ≤ D. In an earlier paper Curtin and Nomura determined the intersection numbers of in terms of D and two complex parameters η and q. We extend their results as follows. Fix any vertex x ∈ X and let T = T (x) denote the corresponding Terwilliger algebra. Let U denote an irreducible T -module with endpoint r and diameter d. We obtain the intersection numbers c i (U ), b i (U ), a i (U ) as rational expressions involving r, d, D, η and q. We show that the isomorphism class of U as a T -module is determined by r and d. We present a recurrence that gives the multiplicities with which the irreducible T -modules appear in the standard module. We compute these multiplicites explicitly for the irreducible T -modules with endpoint at most 3. We prove that the parameter q is real and we show that if is not bipartite, then q > 0 and η is real.

Let Y denote a D-class symmetric association scheme with D ≥ 3, and suppose Y is almostbipartite P-and Q-polynomial. Let x denote a vertex of Y and let T = T (x) denote the corresponding Terwilliger algebra. We prove that any irreducible T -module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T -module is determined by two parameters, the dual endpoint and diameter of W . We find a recurrence which gives the multiplicities with which the irreducible T -modules occur in the standard module. We compute this multiplicity for those irreducible T -modules which have diameter at least D − 3.