Introduction:
A connected graph is a graph in which there is a path between every pair of vertices. In other words, we can reach any vertex from any other vertex in the graph. The number of edges required to ensure that a graph with 10 vertices is connected can be determined using the concept of minimum edges in a connected graph.

Explanation:
To understand why a graph with 10 vertices requires at least 36 edges to be connected, let's consider the minimum number of edges required to connect each vertex.

Minimum edges for a connected graph:
In a connected graph with n vertices, the minimum number of edges required to ensure connectivity is given by the formula:
Minimum edges = n - 1

Applying the formula to the given graph:
In this case, we have 10 vertices, so the minimum number of edges required would be:
Minimum edges = 10 - 1 = 9

However, this formula only provides the minimum number of edges required for connectivity, and it does not guarantee a fully connected graph. To ensure that there is a path between every pair of vertices, we need additional edges.

Extra edges for full connectivity:
To determine the number of extra edges required for full connectivity, we need to consider that each vertex must be connected to every other vertex. So, for each vertex, we need to add edges to the remaining 9 vertices.

Calculating the number of extra edges:
Number of extra edges = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
= 45

Total number of edges:
To find the total number of edges required for a fully connected graph with 10 vertices, we add the minimum edges required to the extra edges:
Total edges = Minimum edges + Number of extra edges
= 9 + 45
= 54

Conclusion:
Therefore, to ensure that a simple graph with 10 vertices is connected, we need a minimum of 9 edges for connectivity and an additional 45 edges for full connectivity. Thus, the correct answer is 36, as it represents the minimum number of edges required for a connected graph.