Let L be subset of {3,4,…} and let Xn,M(L) the number cycles belonging to unicyclic components whose length is in random graph G(n,M). We find limiting distribution subcritical regime M=cn with c<1/2 critical M=n2(1+μn−1/3) μ=O(1). Depending on a condition involving series ∑ℓ∈Lzℓ/(2ℓ), we obtain limit either Poisson or normal as n→∞.

In this paper, we study certain determinants over finite fields. Let $\mathbb{F}_q$ be the field of $q$ elements and let $a_1,a_2,\cdots,a_{q-1}$ all nonzero $\mathbb{F}_q$. $T_q=\left[\frac{1}{a_i^2-a_ia_j+a_j^2}\right]_{1\le i,j\le q-1}$ a matrix We obtain explicit value $\det T_q$. Also, as consequence our result, confirm conjecture posed by Zhi-Wei Sun.