Number of Edges in Random Intersection Graph on Surface of a Sphere

In this article, we consider `$N$’spherical caps of area $4\pi p$ were uniformly distributed over the surface of a unit sphere. We study the random intersection graph $G_N$ constructed by these caps. We prove that for $p = \frac{c}{N^{\al}},:c >0$and$\al >2,$thenumberofedgesingraph$G_N$ follow the Poisson distribution. Also we derive the strong law results for the number of isolated vertices in $G_N$: for $p=\frac{c}{N^{\text{al}}},c>0$ for $\text{al}<1,$ there is no isolated vertex in $G_N$ almost surely i.e., there are atleast $N\mathrm{/}2$ edges in $G_N$ and for $\text{al}>3,$ every vertex in $G_N$ is isolated i.e., there is no edge in edge set ${\text{cE}}_N\mathrm{.}$