$($a$)$ Suppose that there are four processes which are topologically arranged in the form of a four node complete graph. Determine the congestion, dilation and expansion if they are embedded on a $4$ node linear array without any wraparound links.

$($b$)$ If the four processes are embedded on a $4$ node star graph, determine the congestion $,$ dilation and expansion on a $4$ node linear array with wraparound links.

$($c$)$ For each of the above types of topologies of the input processes in sub part $($a$)$ and sub part $($b$),$ determine the expansion if these processes are embedded on a dual core processor.