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 $...

I am trying to figure out what will happen to the uniform matroid $U_{2,m}$ if we remove an element e from it, where e is neither a coloop nor a loop. I am guessing that it will become disconnected but I am not sure from this. A uniform matroid is defined as: If m and n are non-negative integers ...

Are the parallel elements in a matroid just behaving like loops? If so, why? For example, in $U_{2,3}$ if we contract an element what will happen? In $U_{2,2}$ if we contract an element what Will happen? In $U_{2,5},$ if we contract an element what will happen?