Abstract Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of graph is fixed as complement some convex region in plane, and all other vertices are placed at mass center their neighbors, then this results unique embedding, planar. It also follows fairly quickly that minimizes sum squared edge lengths, conditional on face. However, it not clear ho...

We consider the SL ( 2 , Z ) $\operatorname{SL}(2,\mathbb {Z})$ -orbits of primitive n $n$ -squared origamis in stratum H $\mathcal {H}(2)$ . In particular, we 4-valent graphs obtained from action with respect to a generating set size two. prove that, apart orbit for = 3 $n 3$ and one orbits 5 5$ all are non-planar. Specifically, each exhibit K $K_{3,3}$ minor, where is complete bipartite graph...

This work is concerned with consensus control for a class of leader-following multi-agent systems (MASs). The information that each agent received is corrupted by measurement noises. To reduce the impact of noises on consensus, time-varying consensus gains are adopted, based on which consensus protocols are designed. By using the tools of stochastic analysis and algebraic graph theory, a suffic...

A radio mean labeling of a connected graph G is a one to one map f from the vertex set V (G) to the set of natural numbers N such that for each distinct vertices u and v of G, d (u, v)+ ⌈ f(u)+f(v) 2 ⌉ ≥ 1+ diam (G). The radio mean number of f , rmn (f), is the maximum number assigned to any vertex of G.The radio mean number of G, rmn (G) is the minimum value of rmn (f) taken over all radio mea...

This paper is concerned with average-consensus control under directed topologies and random measurement noises. To attenuate the measurement noises, time-varying consensus gains are introduced in the protocol. It is shown that under the protocol designed, all agents’ states converge to a common Gaussian random variable, whose mathematical expectation is just the average of the initial states, a...