POISSON APPROXIMATION FOR THE NUMBER OF ISOLATED CYCLES IN A RANDOM INTERSECTION GRAPH

Given a set V with n vertices and another universal set U with m elements, define a bipartite graph B(n,m, p) with independent vertex sets V and U and edges between v ∈ V and u ∈ U existing independently with probability p. The random intersection graph G(n,m, p) , derived from B(n,m, p), is defined on the vertex set V with vertices v1, v2 ∈ V adjacent if and only if there exist some u ∈ U such that both v1 and v2 are adjacent to u in B(n,m, p). Also define Si be a random subset of U such that each element of Si is adjacent to i ∈ V , in which case two vertices i, j ∈ V are adjacent if and only if Si ∩ Sj 6= φ, and edge set E(G) is define as


Introduction
Given a set V with n vertices and another universal set U with m elements, define a bipartite graph B(n, m, p) with independent vertex sets V and U and edges between v ∈ V and u ∈ U existing independently with probability p.The random intersection graph G(n, m, p) , derived from B(n, m, p), is defined on the vertex set V with vertices v 1 , v 2 ∈ V adjacent if and only if there exist some u ∈ U such that both v 1 and v 2 are adjacent to u in B(n, m, p).Also define S i be a random subset of U such that each element of S i is adjacent to i ∈ V , in which case two vertices i, j ∈ V are adjacent if and only if S i ∩ S j = φ, and edge set E(G) is define as E(G) = {{i, j} : i, j ∈ V, S i ∩ S j = φ}.
The properties of G(n, m, p) were studied in [2,3]  known random graph model G(n, p), in which vertices are made adjacent to each other independently and with probability p, and showed that for a fixed α > 0, the number of elements m is taken to be m = ⌊n α ⌋.In 1999, Karonski, Scheinerman and Singer-Cohen [2] showed that the total variation distance between the distribution of G(n, m, p) and G(n, p) converges to 0 when α > 6 and p is defined appropriately.Without loss of generality we consider the independent set V. For i = 1, 2, 3, ..., n, let Clearly, X is the number of isolated vertices in G(n, m, p).
In 2011, Yilum Shang [6] proved that the distribution function of X can be approximated by Poisson distribution with parameter In 2013, M. Donganont [9] showed the another proof of Poisson approximation for the number of isolated vertices in G(n, m, p) by Stein-Chen and coupling method.The results as the following, Theorem 1.1 (9).Let W be the number of isolated vertices in a random intersection graph G(n, m, p) .
For A ⊆ {0, 1, 2, ...n} and m = ⌊n α ⌋ for some α > 0, we have and When C w = {0, 1, ...., w} Corollary 1.1.[9] Let W be the number of isolated vertices in a random intersection graph G(n, m, p).Let A ⊆ {0, 1, 2, ...n}, m = ⌊n α ⌋ for some α > 0, q = 1 − p, and p = 1 n γ for any γ ∈ R + \ {1} ,then be the set of all possible combinations of k vertices.we note that T k is a tree of order k in G(n, m, p) and say that T k is isolated in G(n, m, p) if there is no edge in G(n, m, p) between a vertex in the tree and the other outside of the tree.
For each i ∈ Γ n,k , we define the indicator random variable there is an isolated tree in G(n, m, p) that spans the vertices i = (i 1 , . . ., i k ), 0 otherwise, and set Then W n,k is the number of isolated trees in G(n, m, p).
In 2013, Mana [10] shows that if m = ⌊n α ⌋ ; α > 0 , then W n,k can be approximated by Poisson approximation with parameter By using Stein-Chen and Coupling Method.The result is following, Theorem 1.2 (10).Let W be the number of isolated trees in a random intersection graph G(n, m, p) .
For A ⊆ {0, 1, 2, ...n} and m = ⌊n α ⌋ for some α > 0, we have and When C w = {0, 1, ...., w} Corollary 1.2.[10] Let W be the number of isolated trees in a random intersection graph G(n, m, p).Let A ⊆ {0, 1, 2, ...n}, m = ⌊n α ⌋ for some α > 0, This work, we use this idea in order to show that the number of isolated cycles can be approximated by Poisson distribution.Now, we define, be the set of all possible combinations of k vertices.we note that C k is a cycle of order k in G(n, m, p) and say that C k is isolated in G(n, m, p) if there is no edge in G(n, m, p) between a vertex in the cycle and the other outside of the cycle.
For each i ∈ Γ n,k , we define the indicator random variable there is an isolated cycle in G(n, m, p) that spans the vertices i = (i 1 , . . ., i k ), 0 otherwise, and set Thus, W n,k is the number of isolated cycles in G(n, m, p).

Stein-Chen and Coupling Method
In 1972, Stein [1] gave a new technique to find a bound in the normal approximation to a distribution of a sum of dependent random variables.His technique was relied instead on the elementary differential equation.In 1975, Chen [4,5] applied Stein's idea to the Poisson case.The central idea of the Stein-Chen method is the difference equation where λ > 0 and A ⊆ N ∪ {0} and I A : N ∪ {0} → R is defined by The equation ( 3) is called Stein's equation for Poisson distribution function and its solution is So far W could be i∈Γ In 1992, Barbour, Holst and Janson [7] constructed coupling random variable W i and used Stein-Chen method to find the bound in Poisson approximation of W .They assumed that for each i the distribution L(W i ) of W i equals to the conditional distribution L(W − X i |X i = 1) and gave the fundamental theorem as follows: Theorem 2.1.If W and W i are defined as above, then where g λ,A := sup In 2006, Santiwipanont and Teerapabolarn [8] proved that for any subset A of {0, 1, . . ., n}, where and In next section, we will use Theorem 2.1 and ( 6) to prove our main result by constructing the random variable W i .

Proof of Main Result
Let W i be the number of isolated cycles of order k in a random intersection graph G(n, m, p) − i, G(n, m, p) − i obtained from G(n, m, p) by dropping the set i ⊆ V and all the edges containing any of these vertices.Then for w 0 ∈ {0, 1, . . ., ⌊ n−k k ⌋}, we have and From ( 7) and ( 8), the distribution of W i equals to the conditional distribution of (W − X i |X i = 1).For i, j ∈ Γ n,k such that i = j, we define the indicator random variable X (i) j and E ij , as follow here is an isolated cycle in G(n, m, p) − i that spans the vertices i = (i 1 , . . ., i k ), 0 otherwise, and 1 if there exists adjacent between i k ∈ i and j l ∈ j, 0 otherwise.