To formulate a linear programming (LP) model that minimizes weekly wages in the research lab under the given constraints, we need to define variables and constraints based on the problem description.

Let's denote:
- \(U\) as the number of undergraduate students.
- \(G\) as the number of graduate students.

The objective function aims to minimize the total weekly wages. Given that undergraduate students earn $100 a week and graduate students earn $300 a week, the objective function can be written as:
\[ \text{Minimize} \quad 100U + 300G \]

Now, let's define the constraints based on the problem description:
1. The lab requires at least 50 students in total:
\[ U + G \geq 50 \]
2. At least 10 of these students must be graduate students:
\[ G \geq 10 \]
3. The number of graduate students should be at least a third the number of undergraduate students:
\[ G \geq \frac{1}{3}U \]

All variables \(U\) and \(G\) should be non-negative since they represent counts of students.

Now, let's translate this model into Gurobi code in Python:

```python
from gurobipy import *

# Create a new model
m = Model("ResearchLab")

# Define the variables
U = m.addVar(vtype=GRB.INTEGER, name="Undergraduate_Students")
G = m.addVar(vtype=GRB.INTEGER, name="Graduate_Students")

# Set the objective function
m.setObjective(100*U + 300*G, GRB.MINIMIZE)

# Add constraints
m.addConstr(U + G >= 50, "Total_Students")
m.addConstr(G >= 10, "Minimum_Graduate_Students")
m.addConstr(G >= (1/3)*U, "Graduate_vs_Undergraduate")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Undergraduate Students: {U.x}")
    print(f"Graduate Students: {G.x}")
    print(f"Total Weekly Wages: ${100*U.x + 300*G.x}")
else:
    print("No optimal solution found")
```