Final answer:
The questions involve concepts of graph theory. A tree on 10 nodes with 11 edges cannot exist because trees must have n-1 edges where n is the number of nodes. Trees cannot have a distinct central and centroid vertex, nor can they have two distinct central and two distinct centroid vertices. A graph with κ=λ=δ=2 can be a cycle graph with at least 3 vertices.
Step-by-step explanation:
The question is asking to verify if certain types of trees with specific properties exist. The concept of trees here is related to graph theory, a branch of mathematics.
- A tree on 10 nodes cannot have 11 edges because in graph theory, a tree with n nodes always has n-1 edges. Hence, a tree with 10 nodes should have 9 edges. This means a tree with 10 nodes and 11 edges cannot exist.
- A tree cannot have exactly one central vertex and one centroid vertex that are distinct if it has at least 3 vertices, because in a tree, the center and centroid are the same or are adjacent to each other. Therefore, no such tree exists.
- A tree cannot have exactly two central vertices and two centroid vertices and all are distinct. When a tree has two of either, they must be adjacent, meaning they cannot all be distinct.
- For a graph G, where κ(G) is the vertex connectivity, λ(G) is the edge connectivity, and δ(G) is the minimum degree of the graph, if all are equal to 2, it suggests that G is a graph without any cut-vertices, has at least two edges connecting any partition, and every vertex has at least two neighbors. An example of such a graph is a cycle graph with at least 3 vertices.