Regular graphs of degree at most four that allow two distinct eigenvalues

For an n×n matrix A, let q(A) be the number of distinct eigenvalues of A. If G is a connected graph on n vertices, let S(G) be the set of all real symmetric n×n matrices A=[aij] such that for i≠j, aij=0 if and only if {i,j} is not an edge of G. Let q(G)=min{q(A):A∈S(G)}. Studying q(G) has become a fundamental sub-problem of the inverse eigenvalue problem for graphs, and characterizing the case for which q(G)=2 has been especially difficult. This paper considers the problem of determining the regular graphs G that satisfy q(G)=2. The resolution is straightforward if the degree of regularity is 1,2, or 3. However, the 4-regular graphs with q(G)=2 are much more difficult to characterize. A connected 4-regular graph has q(G)=2 if and only if either G belongs to a specific infinite class of graphs, or else G is one of fifteen 4-regular graphs whose number of vertices ranges from 5 to 16. This technical result gives rise to several intriguing questions.