QUESTION $1(20$ MARKS$)$

$($a$)$ Given the incidence matrix

$B=\{[1,1,0,0,0,0$

$1,0,1,0,1,0$

$0,0,1,1,0,0$

$0,1,0,1,0]\}$

$000011$

and the weights for each of the edges

$($i$)$ Draw the equivalent graph, G

$($ii$)$ Write the adjacent matrix for graph, G

$($iii$)$ Is the graph G Hamilton? Explain

$($iv$)$ Is the graph Eulerian? Explain

$($v$)$ Find $x(G)$

$($vi$)$ Is it possible to obtain a perfect matching from graph G$?$ Explain.