Diminished Sombor Index, Spectrum and Energy of Comaximal Graphs of Commutative Rings

In this paper we introduce and study the diminished Sombor index and the associated diminished Sombor spectrum of the comaximal graph ({Z}ₙ) of the ring of integers modulo n. The diminished Sombor matrix is defined by replacing each nonzero off-diagonal entry of the adjacency matrix with the normalised weight aligned d (u) ²+d (v) ²d (u) +d (v), aligned where d (u) and d (v) denote the degrees of the adjacent vertices u, v. We establish general bounds for the diminished Sombor index DSO ( ({Z}ₙ) ), characterise the extremal cases, and describe in detail the diminished Sombor spectrum of ({Z}ₙ). In particular, we show that the eigenvalue -1/2 occurs with multiplicity (n) -1, the eigenvalue 0 occurs with multiplicity n- (n) -1-t, and that the remaining t+2 eigenvalues are precisely the eigenvalues of a reduced quotient matrix depending only on n and its prime-power divisors. Using this spectral description we derive explicit formulas and bounds for the diminished Sombor energy of ({Z}ₙ). We analyzed certain special cases viz. prime, prime power, and product of two or three primes in detail, and illustrative numerical examples are provided. Finally, we conclude the paper with some open problems and directions for future research.