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For an $n \times n$ matrix $A$, let $q(A)$ be the number of distincteigenvalues of $A$. If $G$ is a connected graph on $n$ vertices, let$\mathcal{S}(G)$ be the set of all real symmetric $n \times n$ matrices$A=[a_{ij}]$ such that for $i\neq j$, $a_{ij}=0$ if and only if $\{i,j\}$ isnot an edge of $G$. Let $q(G)={\rm min}\{q(A)\,:\,A \in \mathcal{S}(G)\}$.Studying $q(G)$ has become a fundamental sub-problem of the inverse eigenvalueproblem for graphs, and characterizing the case for which $q(G)=2$ has beenespecially difficult. This paper considers the problem of determining theregular graphs $G$ that satisfy $q(G)=2$. The resolution is straightforward ifthe degree of regularity is $1, 2,$ or $3$. However, the $4$-regular graphswith $q(G)=2$ are much more difficult to characterize. A connected $4$-regulargraph has $q(G)=2$ if and only if either $G$ belongs to a specific infiniteclass of graphs, or else $G$ is one of fifteen $4$-regular graphs whose numberof vertices ranges from $5$ to $16$. This technical result gives rise toseveral intriguing questions.

Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues