9. (a) Use pseudocode to describe an algo- rithm for determining the value of a game tree when both players follow a minmax strategy. (b) Suppose that T₁ and T2 are spanning trees of a simple graph G. Moreover, suppose that ₁ is an edge in T₁ that is not in T2. Show that there is an edge 2 in T2 that is not in T₁ such that T₁ remains a spanning tree if ₁ is removed from it and 2 is added to it, and T2 remains a spanning tree if 2 is removed from it and e₁ is added to it. (c) Show that a degree-constrained spanning tree of a simple graph in which each vertex has degree not exceeding 2 2 consists of a single Hamiltonian path in the graph.

the number of spanning trees of a labels c3 ?

Show that any graph with two or more nodes and no self-loops contains two nodes that have equal degrees.

I have a graph with 2021 vertices. It is not a tree but there exists some vertex whose removal makes it a tree. What is the maximum possible number of edges in the original graph?

A school consists of 6 separate buildings, represented by the vertices in the graph. There are paths between some of the buildings as shown. The graph also shows the length in feet of each path. School administrators want to cover some of these paths with roofs so that students will be able to walk between buildings without getting wet when it rains. To minimize cost, they must select paths to be covered such that the total length to be covered is as small as possible. Use Kruskal's algorithm to determine which paths to cover. Also determine the total length of pathways to be covered. OA. Click here to view figure c. OC. Click here to view figure b. A The total length of pathways to be covered is (Type a whole number.) 45 B Find a minimum spanning tree. Choose the correct answer below. 51 34 19 39 25 E 31 29 30 25 F41 (... D 35 O B. Click here to view figure d. O D. Click here to view figure a.

6. (a) Write Prims algorithm for determine the minimum spanning tree of weighted graph. Also determine the spanning tree of following graph using Prim's algorithm. Show the spanning tree after tracing each step of algorithm. 12 e 15 10 11 14 13 8 (b) Represent the following graph using adjacency matrix and adjacency list. 5 6 -2 8 -3 y

3 Show that it is not possible to creak a graph with 9 vertices such that the degree ( a of every vertex is 3.

Let (u, v) be a minimum-weight edge in a connected graph G. Show that (u, v) belongs to some minimum spanning tree of G.