Create a matrix with a given sparsity pattern and whose condition number is low

Question

In Matlab, I need to create a symmetric matrix with a given sparsity pattern and whose condition number is low ($\leq 10$). The matrix is sparse (more than half of its entries are zero).

From what I already know, generally if diagonal entries are larger and if the value of the entries are taken from a function, the condition number will be low.

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Update

As an example of structure, we can consider Jacobian matrices or matrices whose graph is a star.

Answer 1

Is there a reason you are not using sprandsym(n,density,rc) with density$\geq 0.5$ and rc$\in (0.1,\infty)$ (documentation)?