John S. Caughman

Decentralized control of vehicle formations

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.

The Terwilliger algebras of bipartite P - and Q -polynomial schemes

Abstract Let Y denote a D -class symmetric association scheme with D ⩾ 3, and suppose Y is bipartite P - and Q -polynomial. Let T denote the Terwilliger algebra with respect to any vertex x . 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 endpoint and diameter of W . We find a recurrence which gives the multiplicities with which the irreducible T -modules occur in the standard module. Using this recurrence, we produce formulas for the multiplicities of the irreducible T -modules with endpoint at most four.Let Y denote a D-class symmetric association scheme with D > 3, and suppose Y is bipartite Pand Q-polynomial. Let T denote the Terwilliger algebra with respect to any vertex X. 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 7’ on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the irreducible T-modules occur in the standard module. Using this recurrence, we produce formulas for the multiplicities of the irreducible T-modules with endpoint at most four. @ 1999 Elsevier Science B.V. All rights reserved AMS classification: primary 05E30

Spectra of Bipartite P- and Q-Polynomial Association Schemes

Abstract. Let Y=(X,{Ri}0≤i≤D) denote a symmetric association scheme with D≥3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A0, A1,…, AD of the associate matrices, and Q-polynomial with respect to the ordering E0, E1,…,ED of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)–(iv).(i) (ii) D is even, and (iii) θ*0>θ0, and (iv) θ*0>θ0, D is odd, and

The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme

Let Y denote a D-class symmetric association scheme with D>=3, and suppose Y is almost-bipartite 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.

The last subconstituent of a bipartite Q -polynomial distance-regular graph

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 3. Fix any vertex x and let ΓD = ΓD(x) denote the set of vertices at distance D from x. Let ΓD2 = ΓD2(x) denote the graph with vertex set ΓD, and edge set consisting of all pairs of vertices in ΓD which are at distance 2 in Γ. In this paper, we assume Γ is Q-polynomial and show ΓD2 is distance-regular and Q-polynomial. We compute the intersection numbers of ΓD2 from the intersection numbers of Γ. To obtain our results, we use a characterization of the Q-polynomial property due to Terwilliger.

Intersection numbers of bipartite distance-regular graphs

Abstract Let l denote a bipartite distance-regular graph with diameter d ⩾2, and intersection numbers 1= c 1 , c 2 ,…, c d = k . For any integer i (1⩽ i ⩽ d −1), we find a collection of lower bounds for the quantity c i +1 −1− c i ( μ −1) where μ := c 2 . Our results imply that if kc i ( (μ−1)(μ−2)(c i −c i−1 −1) 2 +1) then c i +1⩾ c i ( μ −1)+1.

Bipartite Q -Polynomial Distance-Regular Graphs

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 an integer at least 2. We obtain the above result using the Terwilliger algebra of C.Abstract.Let Γ denote a bipartite distance-regular graph with diameter D≥12. We show Γ is Q-polynomial if and only if one of the following (i)–(iv) holds: (i) Γ is the ordinary 2D-cycle. (ii) Γ is the Hamming cube H(D,2). (iii) Γ is the antipodal quotient of H(2D,2). (iv) The intersection numbers of Γ satisfy where q is an integer at least 2. We obtain the above result using the Terwilliger algebra of Γ.

Reversible Function Synthesis of Large Reversible Functions with No Ancillary Bits Using Covering Set Partitions

This paper presents a synthesis algorithm, Covering Set Partitions (CSP), for reversible binary functions with no ancillary (garbage) bits. Existing algorithms are constrained to functions of small number of variables because they store the entire truth table of 2n terms in memory or require a huge amount of time to yield results because they must calculate all possible permutations of an input vector. In contrast, the CSP algorithm harnesses the natural mathematical properties of binary numbers, partially ordered sets and covering graph theory, to construct input vectors which are guaranteed to produce valid results. A randomly selected subset of all valid input vectors are processed where the best input vector sequence wins. The CSP algorithm is capable of synthesizing functions of large number of variables (30 bits) in a reasonable amount of time.

Set Partitions and the Multiplication Principle

Abstract To further understand student thinking in the context of combinatorial enumeration, we examine student work on a problem involving set partitions. In this context, we note some key features of the multiplication principle that were often not attended to by students. We also share a productive way of thinking that emerged for several students who were then able to resolve the issues in question. We conclude with pedagogical implications.

The Parameters of Bipartite Q -polynomial Distance-Regular Graphs

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, ..., θD is a Q-polynomial ordering of the eigenvalues of Γ. This sequence is known to satisfy the recurrence θi − 1 − βθi + θi + 1 = 0 (0 > i > D), for some real scalar β. Let q denote a complex scalar such that q + q−1 = β. Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D ≥ 4. Moreover, if D = 3, then q is not real if and only if θ1 is the second largest eigenvalue and the pair (μ, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.