Construct a graph with the properties specified below. If no such graphs exist, explain why.

$($a$)$ Six nodes, exactly three nodes of even degree.

$($b$)$ A graph on $7$ nodes where each node has degree exactly $3.$

$($c$)$ Self complementary graph on $7$ nodes.

$($d$)$ Six nodes, $16$ edges.

$($e$)$ Eight nodes, exactly two nodes of degree $3,$ all other nodes of degree $2$ or $4$ but no path connecting degree $3$ vertices.

$($f$)10$ nodes where each node is of degree $1.$