The largest integer m such that every simple connected graph with n vertices and n edges contains at least m different spanning trees is 3 (option C).

Explanation:
A spanning tree of a graph is a connected subgraph that includes all the vertices of the original graph and contains no cycles. In other words, it is a tree that spans all the vertices of the graph.

To determine the largest integer m that guarantees at least m different spanning trees in any simple connected graph with n vertices and n edges, we can analyze the properties of spanning trees.

1. Minimum number of edges in a spanning tree:
A simple connected graph with n vertices will have at least n-1 edges in any spanning tree. This is a property of trees.

2. Maximum number of edges in a spanning tree:
A simple connected graph with n vertices and n edges will have exactly n-1 edges in any spanning tree. This is because adding any additional edge will create a cycle in the tree.

Based on these properties, we can determine the minimum and maximum number of edges in a spanning tree for a simple connected graph with n vertices and n edges.

- Minimum number of edges in a spanning tree: n-1
- Maximum number of edges in a spanning tree: n-1

Since a spanning tree is defined by its set of edges, the number of different spanning trees can be determined by the number of distinct sets of edges that satisfy the properties of a spanning tree. Therefore, the number of different spanning trees must be greater than or equal to the number of different sets of n-1 edges.

- Number of different sets of n-1 edges: C(n, n-1) = n

Thus, the largest integer m that guarantees at least m different spanning trees in any simple connected graph with n vertices and n edges is 3, as there can be at most n different sets of n-1 edges.

Therefore, the correct answer is option C) 3.