text: Random @math -intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random @math -intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least @math items (the graph is called a random intersection graph when @math ). A special case of particular interest is a uniform random @math -intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another important case is a binomial random @math -intersection graph, where each item from a pool is independently assigned to each vertex with the same probability. Both models have found numerous applications thus far including cryptanalysis, and the modeling of recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random @math -intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random @math -intersection graphs). In this paper, for binomial uniform random @math -intersection graphs, we present results related to @math -connectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zero-one laws for the following three properties: (i) @math -vertex-connectivity, (ii) @math -edge-connectivity and (iii) the property of minimum vertex degree being at least @math .
  @cite_17: Let W1,…,Wn be independent random subsets of [m]= 1,…,m . Assuming that each Wi is uniformly distributed in the class of d-subsets of [m] we study the uniform random intersection graph Gs(n,m,d) on the vertex set W1,…Wn , defined by the adjacency relation: Wi∼Wj whenever |Wi∩Wj|≧s. For even n we show that as n,m→∞ the edge density threshold for the property that Gs(n,m,d) contains a perfect matching is asymptotically the same as that for Gs(n,m,d) being connected.
summary: For graph @math , Bloznelis and uczak @cite_17 recently consider the following three properties: (i) the graph has a minimum vertex degree at least @math ; (ii) the graph has a perfect matching; (iii) the graph is connected. They present that when @math is even, for each of these three properties, its probability converges to @math as @math , under conditions @math ( @math is the edge probability), @math and @math for some @math .