2016/08/30 by Hirasaka, Mitsugu, Oh, Semin
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1608.08508
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≥ 1an n-s is well-defined where an 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.