Aγ Eigenvalues of Zero Divisor Graph of Integer Modulo and Von Neumann Regular Rings

Abstract

The Aγ matrix of a graph G is determined by Aγ(G)=(1−γ)A(G)+γD(G), where 0≤γ≤1, A(G) and D(G) are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the Aγ matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more γ adjacency-type matrices. In this paper, we obtain the Aγ eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs.