Question $4$

Consider the graph $R$ in Figure Q$4$ below.

$($a$)$ Find an independent set of vertices in $R$ that has the largest number of vertices. Calculate

the independence number, $(R),$ of graph $R.$

$($b$)$ Find a matching in $R$ that has the largest number of edges. What is the matching number,

${}^{\text{'}}(R),$ of graph $R?$

$($c$)$ Find a minimum vertex cover of $R.$ What is the vertex covering number, $(R),$ of $R?$

$($d$)$ Find a minimum edge cover of $R.$ What is the edge covering number, ${}^{\text{'}}(R),$ of $R?$

$($e$)$ Let $S$ be a simple graph with $37$ vertices such that maximum degree of a vertex is $(S)=$

$5$ and the minimum degree of a vertex is $()=1.$ Prove or disprove that $$ has a matching

with $7$ or more edges.