To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints using these variables.

Let's define:
- $x_1$ as the number of undergraduate students.
- $x_2$ as the number of graduate students.

The objective is to minimize the total weekly wages, which can be represented as $100x_1 + 300x_2$, since each undergraduate student earns $100 a week and each graduate student earns $300 a week.

The constraints based on the problem description are:
1. The lab requires at least 50 students in total: $x_1 + x_2 \geq 50$.
2. At least 10 of the students must be graduate students: $x_2 \geq 10$.
3. The number of graduate students should be at least a third the number of undergraduate students: $x_2 \geq \frac{1}{3}x_1$.

All variables ($x_1, x_2$) must be non-negative since they represent quantities of students.

The symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'number of undergraduate students'), ('x2', 'number of graduate students')],
  'objective_function': '100*x1 + 300*x2',
  'constraints': ['x1 + x2 >= 50', 'x2 >= 10', 'x2 >= (1/3)*x1', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this linear programming problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="undergraduate_students", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="graduate_students", lb=0)

# Set the objective function
m.setObjective(100*x1 + 300*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(x1 + x2 >= 50, "total_students")
m.addConstr(x2 >= 10, "minimum_graduate_students")
m.addConstr(x2 >= (1/3)*x1, "graduate_to_undergraduate_ratio")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Undergraduate students: {x1.x}")
    print(f"Graduate students: {x2.x}")
    print(f"Total cost: {m.objVal}")
else:
    print("No optimal solution found")
```