91影视

A minimal spanning tree is indeed an edge-weighted graph with a weight that is equal to which is less than the value about any other spanning tree.

a)

Minimum spanning tree cost:

The number of edges for a minimal spanning tree may be calculated using the formula below.:

$\mathrm{Number}\mathrm{of}\mathrm{edges}=\mathrm{Number}\mathrm{of}\mathrm{vertices}鈥�1$

In Equation, replace$"\mathrm{number}\mathrm{of}\mathrm{vertices}=8"$with the value.

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{edges}=8鈥�1 \\ =7 \end{gathered}$$

The following are the steps to determining the least spanning tree:

Begin with the graph's vertices " A ."

鈥� Each vertices "A " has two angles with such a load of 6 vertices point A to point B and 1 vertices from to .

鈥� From A through E vertices, the minimum weight is 1 as indicated below:

Next, starts with vertex 鈥�B 鈥�.

鈥� The vertex "B " has four edges with weights of 6 vertices from B to A ,5 vertices from B to C ,2 vertices from B to E , and 2 vertices from B to F . From B to E vertices, the minimum weights are 2; from B to F vertices, the minimum weights are 2. Take a look at the initial minimal weight value. As a result, the minimal weight 2 from B to E vertices are as follows:

Then comes the vertex " C ."

鈥� Each vertices " C " has four edges having weights of 5 vertices through C to B ,6 vertices as C to D,5 vertices from C to F, and 4 vertices from C to G.

鈥� From C through G vertices, the lower limit is 4 as indicated below:

Then comes the vertex "D ."

鈥� Each vertices "D" has four edges having weights of 5 vertices through D to B,6 vertices as D to D ,5 vertices from D to F, and 4 vertices from D to G.

鈥� From D through G vertices, the lower limit is 4.

Then comes the vertex " E ."

鈥� The vertex "E " has three edges with weights of one from E to A , two from E to B, and one from E to F vertices.

鈥� The first and second weights have already been drawn.

From E through F vertices, the minimum weight is 1 as indicated below:

Next, begin with vertex "F ."

鈥� The vertex "F " has four edges, each with a weight of one from F to E , two from F to B , five between F and C , and three from F to G .

鈥� The first and second weights have already been drawn.

From F through G vertices, the minimum weight is 3 as indicated below:

The following vertex is "$G$."

鈥� The vertex "G" has four edges with weights of 3 between G and F vertices,4 between G and C vertices,5 between G and D vertices, and 3 between G and H vertices.

鈥� The third and fourth weights have already been drawn.

o From G through H vertices, the minimum weight is 3 as illustrated below:

Finally, the minimum spanning tree is shown below:

The cost of the minimum spanning tree is the sum of all the weighted edges.

Therefore, the cost of the minimum spanning tree: 19

b)

In the given graph, there are two possibilities of the minimum spanning tree. That is, we can take the weight 2 from B to E or weight 2 fromEtoF .

One possible minimum spanning tree with weight$2fromBtoE$and is given below:

Another possible minimum spanning tree with weight$2fromEtoF$and is given below:

Therefore, the number of minimum spanning tree in this graph is 2.

c)

The minimum spanning tree using the Kruskal鈥檚 algorithm is given below:

In this MST,

鈥� There are 7 edges are included into the minimum spanning tree such as,

$AE,EF,BE,FG,GH,CG,andGD.$

$$\begin{gathered} 鈥�For\mathrm{AE}edge,thecutsare\left\{A,B,C,D\right\}and\left\{E,F,G,H\right\}. \\ 鈥�ForEFedge,thecutsare\left\{A,B,C,D,E\right\}and\left\{F,G,H\right\}. \\ 鈥�ForBEedge,thecutsare\left\{A,E,F,G,H\right\}and\left\{B,C,D\right\}. \\ 鈥�ForFGedge,thecutsare\left\{A,B,E\right\}and\left\{C,D,F,G,H\right\}. \\ 鈥�ForGHedge,thecutsare\left\{A,B,E,F,G\right\}and\left\{C,D,H\right\}. \\ 鈥�ForCGedge,thecutsare\left\{A,B,E,F,G,H\right\}and\left\{C,D\right\}. \\ 鈥�ForGDedge,thecutsare\left\{A,B,C,E,F,G,H\right\}and\left\{D\right\}. \end{gathered}$$