John S. Caughman IV

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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 inneighbors in G. We prove that a necessary and(More)
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(More)
Let Y denote a D-class symmetric association scheme with D ≥ 3, and suppose Y is almostbipartite Pand 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(More)
Let Y = (X, {Ri}0≤i≤D) denote a symmetric association scheme, and assume that Y is Q-polynomial with respect to an ordering E0, ..., ED of the primitive idempotents. In [1, p.205], Bannai and Ito conjectured that the associated sequence of multiplicities mi (0 ≤ i ≤ D) of Y is unimodal. We prove that if Y is dual-thin in the sense of Terwilliger, the(More)
Let C denote a bipartite distance-regular graph with diameter D 12. We show C is Q-polynomial if and only if one of the following (i)–(iv) holds: (i) C is the ordinary 2D-cycle. (ii) C is the Hamming cube HðD; 2Þ. (iii) C is the antipodal quotient of Hð2D; 2Þ. (iv) The intersection numbers of C satisfy ci 1⁄4 qi 1 q 1 ; bi 1⁄4 qD qi q 1 ð0 i DÞ; where q is(More)