This paper examines cluster consensus for multi-agent systems on matrix-weighted switching networks. Necessary and/or sufficient conditions under which can be achieved are obtained, as well quantitative characterization of the steady-state consensus. Specifically, when underlying network switches amongst a finite number networks, necessary condition networks is derived; moreover, it shown that ...

In CAD/CAM modeling, objects are represented using the Boundary Representation (ANSI Brep) model. Detection of possible intersection between objects can be based on the objects’ boundaries (i.e., triangulated surfaces), and computed using triangle-triangle intersection. Usually only a cross intersection algorithm is needed; however, it is beneficial to have a single robust and fast intersection...

A theorem of A. Schrijver asserts that a d–regular bipartite graph on 2n vertices has at least ( (d− 1)d−1 dd−2 )n perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive in...

Let L(Qt) denote the number of linear extensions of the t-dimensional Boolean lattice Qt. We use the entropy method of Kahn to show that log(L(Qt)) 2t = log ( t bt/2c ) − 3 2 log e + o(1), where the logarithms are base 2. We also find the exact maximum number of linear extensions of a d-regular bipartite order on n elements, in the case when n is a multiple of 2d.

An optimization problem arising in the design of optical networks is shown here to be abstracted by the following model of covering the edges of a bipartite graph with a minimum number of 4-cycles, or K2,2: Given a bipartite graph G = (L,R,E) over the node set L ∪ R with E ⊆ {[u, v] : u ∈ L, v ∈ R}, and the implicit collection of all fournode cycles in the complete bipartite graph over L∪R. The...

In this paper, we prove the following two theorems. Theorem 1 Let 0 denote a distance-regular graph with diameter d ≥ 3. Suppose E and F are primitive idempotents of 0, with cosine sequences σ0, σ1, . . . , σd and ρ0, ρ1, . . . , ρd , respectively. Then the following are equivalent. (i) The entry-wise product E ◦ F is a scalar multiple of a primitive idempotent of 0. (ii) There exists a real nu...

Definition (Maximal size complete bipartite induced subgraph). β(G) := size of maximal complete bipartite induced subgraph of G. Definition (Minimal bipartite decomposion number). τ(G) := minimal number of complete edge disjoint covering bipartite subgraphs of G. Definition (Minimal nontrivial bipartite decomposion number). τ (G) := minimal number of complete edge disjoint covering nontrivial (...

On the punctured torus the number of essential self-intersections of a homotopy class of closed curves is bounded (sharply) by a quadratic function of its combinatorial length (the number of letters required for its minimal description in terms of the two generators of the fundamental group and their inverses). We show that if a homotopy class has combinatorial length L, then its number of esse...