A subset D of vertices of a graph G is a dominating set if for each u ∈

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A set D of vertices of a graph G is a dominating set if each vertex of V (G) \ D, is adjacent to some vertex of D. The domination number of G, γ(G), is the minimum cardinality of a dominating set of G. A graph G is called domination vertex critical, or just γ-critical if removal of any vertex decreases the domination number. A graph G is called domination vertex stable, or just γ-stable, if removal of any vertex does not decrease the domination number. For an even integer n ≥ 2 and 1 ≤ ∆ ≤ ⌊log 2 n⌋, a Knödel graph W ∆,n is a ∆-regular bipartite graph of even order n, with vertices (i, j), for i = 1, 2 and 0 ≤ j ≤ n/2 − 1, where for every j, 0 ≤ j ≤ n/2 − 1, there is an edge between vertex (1, j) and every vertex (2, (j + 2 k − 1) mod (n/2)), for k = 0, 1, · · · , ∆−1. In this paper, we study the domination criticality and domination stability of Knödel graphs. We characterize the 3-regular and 4-regular Knödel graphs by γcriticality or γ-stability.

In this paper a new approach for the stability analysis of Takagi-Sugeno (T-S) fuzzy systems is proposed. Because of low numbers of Linear Matrix Inequality (LMls) or Bilinear Matrix Inequality (BMls) and independence of this method of finding only one symmetric positive definite matrix makes this method highly applicable and has less computation. Such that for a conventional method which is used for stability analysis, computation length is decreased drastically. Also, the conditions on the shape of fuzzy sets have high latitude which make this method more useful. Based on the properties analysis of the Takagi-Sugeno (T-S) fuzzy systems with two-overlapped fuzzy sets, at first we introduce a method that decreases LMls (BMls) and computations. Then based on this approach a typical theorem will be introduced. To demonstrate the new approach, we apply it to a Mass-Spring-Damper system.

Let G be a graph and k be a positive integer. A vertex set D is called a k-distance dominating set of G if each vertex of G is either in D or at a maximum distance k from some vertex of D. k-distance domination number of G is the minimum cardinality among all k-distance dominating sets of G. In this note we give upper bounds on the k-distance domination number of a connected bipartite graph, and improve some results have been given like Theorems 2.1 and 2.7 in [Tian and Xu, A note on distance domination of graphs,