  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 $\al >2,$ the number of edges in graph $G_N$
follow the Poisson distribution. Also we derive the strong law results for the
number of isolated vertices in $G_N$: for $p = \frac{c}{N^{\al}},\:c >0$ for
$\al < 1,$ there is no isolated vertex in $G_N$ almost surely i.e., there are
atleast $N/2$ edges in $G_N$ and for $\al >3,$ every vertex in $G_N$ is
isolated i.e., there is no edge in edge set $\cE_N.$
