Problem $2$ Prim's Min. Spanning Trees Algorithm

$3$ points

$($a$)$ Demonstrate Prim's algorithm $($with vertex $v_{10}$ as the start node$)$ for the Petersen graph shown in Figure $1.(1\mathrm{.}5$ points$)$

$($b$)$ Prove the correctness of Prim's algorithm formally, i$.$e$.$ show that Prim's algorithm always returns the minimum spanning tree of any given graph and some start node within the graph. $(2\mathrm{.}5$ points$)$

Figure $1$: Petersen Graph