DISTANCE AND ECCENTRICITY SPECTRA OF COMPRESSED ZERO-DIVISOR GRAPHS OVER ∏_(k=1)^n▒F_k , WHERE F_k are any fields and 2≤n≤5

Abstract

Compressed zero–divisor graphs provide a useful connection between ring theory and spectral graph theory, allowing algebraic structures to be studied through graph–theoretic and matrix–based methods. In this work, we examine the compressed zero–divisor graphs corresponding to the finite direct product rings  for  , where . For each case, the associated distance matrix and eccentricity matrix of the compressed zero–divisor graph are constructed explicitly. Their spectral properties are then investigated by determining the characteristic polynomials, eigenvalues, spectral radii, and the corresponding matrix energies. The study shows that as the number of components in the ring increases, the spectral behaviour of both matrices changes noticeably, reflecting the growing structural complexity of the underlying graphs. Overall, the results illustrate that the spectral invariants derived from distance and eccentricity matrices provide meaningful insight into the structure of compressed zero–divisor graphs arising from different ring constructions. This work highlights the close relationship between algebraic graph theory and spectral matrix analysis.