Pascal-type matrices and eigenvalues of zero-divisor graphs of reduced rings

The Pascal-type matrices play a significant role in combinatorial matrix theory and probability. This paper demonstrates an interconnection of certain Pascal-type matrices P and Q with the spectrum of the zero-divisor graph Γ(Rn) of the reduced ring Rn:=Fm×Fm×…×Fm (with n terms), where Fm denotes a finite field with m elements (m being a power of a prime). The graph Γ(Rn) serves as a generalization of the well-known Boolean graph. This paper builds on the spectral study of the Boolean graph by LaGrange and others, to characterize the spectrum of Γ(Rn). The matrices P and Q are utilized to study the spectrum of Γ(Rn). These matrices are both of order n−1, significantly smaller than the order mn−(m−1)n−1 of the adjacency matrix of Γ(Rn). By using Melham and Cooper's work on the eigenvalues of a certain matrix of binomial coefficients, the eigenvalues and the eigenvectors of Q are determined. Further, employing Cauchy's interlacing theorem, it is established that the eigenvalues of the matrices P and −Q, alongside the eigenvalue 0, completely determine all the eigenvalues of the graph Γ(Rn). Moreover, the multiplicity of each of these eigenvalues is determined.