# The values of $m,n$ that makes $U_{m,n}$ graphic.

I am trying to find the values of $$m,n$$ that makes $$U_{m,n}$$ graphic. I am guessing that it is only graphic if $$n = m + 1,$$ I tried $$U_{2,4},U_{2,5},U_{3,6}, U_{2,3},U_{4,5}$$ and I think I am correct. Am I correct or I am missing something?

EDIT:

Here is the definition of $$U_{m,n}:$$

Let $$m$$ and $$n$$ be nonnegative integers with $$m \leq n.$$ Let $$E$$ be an $$n$$-element set and $$\mathcal{B}$$ be the collection of $$m$$ element subsets of $$E.$$ Then $$\mathcal{B}$$ is the set of bases of a matroid on $$E$$ and we denote this matroid by $$U_{m,n}.$$

• It would be helpful to say what $U_{m,n}$ denotes... Sep 24, 2023 at 21:27
• @Math1000 sorry about that I edited my post Sep 24, 2023 at 23:30
• You gave the definition of a regular matroid, but to my knowedge $U_{m,n}$ generally denotes a uniform matroid. It is still confusing. Sep 25, 2023 at 5:38
• @Math1000 you are correct I am so sorry I will edit my question Sep 25, 2023 at 16:09

Note that $$U_{1,n}$$ is the graphic matroid of an $$n$$-edge dipole graph, and the same is true for the corresponding duals - $$U_{n-1,n}$$ and the $$n$$-edge cycle graph.
The same is true for $$U_{0,n}$$, the graphic matroid of a graph with $$n$$ self-loops, and $$U_{n,n}$$, the graphic matroid of an $$n$$-edge forest.
Any other uniform matroid $$U_{m,n}$$ with $$1 contains $$U_{2,4}$$ as a minor and therefore is not graphic.