What is the largest integer m such that every simple connected graph with n…

Answer: 3

Key idea: A connected simple graph with n vertices and n edges is unicyclic (it has exactly one cycle).

• Compute cycle count: For a connected graph, the number of independent cycles equals E - V + 1. Here E = n and V = n, so E - V + 1 = 1, so there is exactly one cycle.

• Let k be the length of that unique cycle. In a simple graph k ≥ 3. Every spanning tree must include all edges not on the cycle (those are bridges) and must be formed by deleting exactly one edge from the cycle. Hence the number of spanning trees equals k.

• Therefore the minimum possible number of spanning trees across all such graphs equals the minimal possible cycle length, which is 3. So every such graph has at least 3 spanning trees.

Example achieving the minimum: Take a triangle (3-cycle) and attach the remaining n-3 vertices as trees (pendant branches) to any vertices of the triangle. The graph is connected and has exactly one cycle of length 3, so it has exactly 3 spanning trees.

Conclusion: 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.