answer:To determine whether the graph is connected or not, we can use the fact that a connected graph with n vertices must have at least n-1 edges. In this case, n = 8, so the minimum number of edges required for the graph to be connected is 8 - 1 = 7 edges. Since the given graph has 10 edges, which is more than the minimum required (7 edges), it is possible for the graph to be connected. However, this does not guarantee that the graph is connected, as the extra edges could form cycles within the graph. To determine if the graph is connected or not, we would need more information about the specific arrangement of the vertices and edges.

answer:To determine the connectivity of the graph, we need to find the minimum number of vertices that need to be removed to disconnect the graph. In this case, we can see that removing vertex E will disconnect the graph into two separate components: ``` A---B C | | D F ``` Thus, the connectivity of the graph is 1, as removing a single vertex (E) is enough to disconnect the graph.

answer:To determine whether the graph is connected or not, we can use the fact that a simple undirected graph with n vertices is connected if it has at least (n-1) edges and the number of edges does not exceed n(n-1)/2. In this case, the graph has 10 vertices and 15 edges. The minimum number of edges required for the graph to be connected is (10-1) = 9 edges. Since 15 > 9, the graph could be connected. However, we need to check if the number of edges does not exceed the maximum possible number of edges for a simple undirected graph with 10 vertices, which is 10(10-1)/2 = 45 edges. Since 15 < 45, the graph is connected. Therefore, the given graph with 10 vertices and 15 edges is connected. There is only one connected component, and its size is equal to the total number of vertices, which is 10.

answer:To determine whether the graph G is connected or not, we will use the fact that a graph with n vertices is connected if and only if it has at least n-1 edges and the sum of the degrees of all vertices is equal to twice the number of edges. In our case, we have n = 9 vertices, so the minimum number of edges required for the graph to be connected is n-1 = 8 edges. Since our graph has 12 edges, which is greater than the minimum required, it is possible for the graph to be connected. Now, let's consider the sum of the degrees of all vertices. Since each edge contributes to the degree of two vertices, the sum of the degrees of all vertices should be equal to 2 * 12 = 24. Let's assume that the graph G is connected. In this case, we can construct a spanning tree of G, which is a connected subgraph that includes all vertices and has no cycles. A spanning tree with 9 vertices will have 9-1 = 8 edges. Now, let's remove the 8 edges of the spanning tree from the graph G. We are left with 12 - 8 = 4 edges. Since the graph is connected, these 4 edges must connect pairs of vertices that are already connected in the spanning tree. This means that adding these 4 edges will create cycles in the graph. However, adding 4 edges to a graph with 9 vertices can create at most 4 cycles. This is because each cycle requires at least 3 vertices, and we have only 9 vertices in total. Therefore, the maximum number of cycles that can be created by adding 4 edges is 4. Since we have shown that it is possible for the graph G to be connected and have 12 edges, we can conclude that the graph G is connected.