Final answer:
In mathematics, it is not possible to construct a tree with 10 nodes and 11 edges. A tree cannot have both a central vertex and a centroid vertex. Similarly, a tree cannot have multiple central and centroid vertices. However, it is possible to construct a graph with a minimum vertex cut, minimum edge cut, and minimum degree equal to 2. In a forest, the number of edges can be calculated as n - k, and a graph with no cycles and n - 1 edges is a tree. A full binary tree on n vertices has n - 1 internal vertices, not n - 1/2 as stated.
Step-by-step explanation:
(a) A tree with 10 nodes must have 9 edges. Since the number of edges in a tree is always one less than the number of nodes, it is not possible to construct a tree with 10 nodes and 11 edges.
(b) A tree cannot have both a central vertex and a centroid vertex. The central vertex is the vertex that has the shortest maximum distance to all other vertices, while the centroid vertex is the vertex that, if removed, would divide the tree into subtrees of equal size. These two concepts are mutually exclusive, so it is not possible to construct a tree with exactly one central vertex and one centroid vertex.
(c) Similar to part (b), a tree cannot have both two central vertices and two centroid vertices. Therefore, it is not possible to construct a tree with exactly two central vertices and two centroid vertices.
(d) To create a graph G such that κ(G)=λ(G)=δ(G)=2, we need a graph where the minimum vertex cut, minimum edge cut, and minimum degree are all equal to 2. One example of such a graph is two disjoint edges, with each edge having two nodes.
(a) True. For a forest with n vertices and k components, the number of edges will be n - k.
(b) True. A graph on n nodes with no cycles and n - 1 edges must be a tree, as a tree is defined as a connected acyclic graph with n - 1 edges.
(c) False. A full binary tree on n vertices will have exactly n - 1 internal vertices and n + 1 leaves. The number of internal vertices is not equal to n - 1/2.