I am trying to find all the matroids having no minors isomorphic to $U_{0,1}$ and $U_{1,2}.$
I know that the matroid $U_{0,n}$ is a matroid of rank zero with n edges and so it is just a vertex with n loops on it. And so it is graphic. And I know that every minor of a graphic matroid is graphic. So, I am guessing that the answer to the first part is all non graphic matroids have no minors isomorphic to $U_{0,1}.$
For $U_{1,2}$.I know that $U_{1,n}$ is a matroid with rank 1 and n edges and I drew it for different n's and I found that its graph is 2 vertices with n-parallel edges so I am guessing also it is graphic and so I am guessing that the answer to the second part is all non graphic matroids have no minors isomorphic to $U_{1,2}.$
Are my guesses correct? If so, is there a way to formalize them in a better way or is there a wider sets of matroids that do not have any of them as minors? If my guesses are incorrect, may I know the correct answer please?
A famous theorem in matroid theory, due to Tutte, characterizes graphic matroids in terms of a set $S$ of excluded minors, meaning a matroid $M$ is graphic if and only if it has no minor in $S$. The set is
$$ S = \{U_{2,4}, F_7, F_7^*, M^*(K_5), M^*(K_{3,3})\}. $$ At a high level, no $U_{2,4}$ minor ensures the matroid is binary, and no $U_{2,4}, F_7, F_7^*$ minor ensures the matroid is regular (which graphic matroids are). The last two exclusion are what really gets at graphicness in some sense. You may recognize them as the dual matroids of the exuded graphs for planarity, due to Kuratowski and Wagner. I suggest you read more about this in Oxley's book, where he goes over this in great detail.
This tells us that matroids without $U_{0,1}$ and $U_{1,2}$ can be both graphic and not, since neither contain a matroid in $S$. You are correct in that $U_{0,1}$ and $U_{1,2}$ are graphic, and the reason for it also correct. $U_{0,1}$ is a loop, and $U_{1,2}$ is a pair of parallel elements. You are asking what class $\mathcal{C}$ of matroids that do not contain a loop or a parallel pair as minors. Observe that contracting an element of a parallel pair yields a loop, or in other words, $U_{0,1}\cong U_{1,2}/\{e\}$ for any element $e$ in $U_{1,2}$. Thus, $U_{0,1}$ is a minor of $U_{1,2}$, implying that $\mathcal{C}$ is in fact the same as the class with no $U_{0,1}$ minor.
Claim. The class $\mathcal{C}$ of matroids with no $U_{0,1}$ minor is the class $\{U_{n,n}:n\in \mathbb{N}\}$.
Proof. Let $\mathcal{C'}=\{U_{n,n}:n\in \mathbb{N}\}$. Now, $\mathcal{C'}$ is exactly the class of matroids $M$ where all subset are of $E(M)$ are independent. You can verify that these matroids have no $U_{0,1}$ minor by recalling the formulas for minors of uniform matroids (contraction, deletion). Suppose a matroid is not in $\mathcal{C'}$, then it contains a dependent set, so a circuit, say $C$. If $|C|=1$, then $C$ is a loop, and we are done. Otherwise, $|C|\geq 2$. Consider $T\subseteq C$ such that $|C\backslash T|=1$, and let $C\backslash T = \{e\}$. Now $$ r(\{e\})_{M/T} = r_M(\{e\} \cup T)-r_M(T) = r_M(C)-r_M(T) = 0, $$ so $r(\{e\}) = 0$, i.e., $e$ is a loop, so $M/T\backslash(E(M)-C)\cong U_{0,1}$, as desired.$\square$