The number of ideals of [formula omitted] containing x(x − α)(x − β) with given index.

It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let B denote an integral square matrix and 〈 B 〉 denote the subring of the full matrix ring generated by B . Then 〈 B 〉 is a free Z -module of finite rank, which guarantees that there are only finitely many ideals of 〈 B 〉 with given finite index. Thus, the formal Dirichlet series ζ 〈 B 〉 ( s ) = ∑ n ≥ 1 a n n − s is well-defined where a n is the number of ideals of 〈 B 〉 with index n . In this article we aim to find an explicit form of ζ 〈 B 〉 ( s ) when B has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, 〈 B 〉 is isomorphic to Z [ x ] / m ( x ) Z [ x ] where m ( x ) is the minimal polynomial of B over Q , and Z [ x ] / m ( x ) Z [ x ] is isomorphic to Z [ x ] / m ( x + γ ) Z [ x ] for each γ ∈ Z . Thus, the problem is reduced to counting the number of ideals of Z [ x ] / x ( x − α ) ( x − β ) Z [ x ] with given finite index where 0 , α and β are distinct integers. [ABSTRACT FROM AUTHOR]