On commuting graphs obtained from classes of completely primary finite rings

Abstract

We study the commuting graph G(R) of a finite non-commutative ring R, where vertices are the non-central elements and edges connect distinct commuting elements. The focus is on the class C of completely primary (finite local) rings R for which the Jacobson radical J satisfies J^3 = 0 and J^2 is contained in Z(R), that is, J^2 is central, while the residue field R/J is the prime field Fp. For every R in C, we prove that R is a CC-ring, meaning that the centralizer of each non-central element is commutative. Consequently, G(R) is a disjoint union of cliques. Let d = dimFp(J) and e = dimFp(J^2). We show that a non-commutative ring exists only when d = e + 2, in which case G(R) consists of N = p + 1 cliques, each of size m = p^(e+1)(p - 1). Using this decomposition, we compute several graph invariants. In particular, the spectrum is integral, the energy is given by E = 2(p^(d+1) - p^(e+1) - p - 1), and the genus satisfies gamma = N.gamma(Km). We further characterize planarity and prove that G(R) is planar if and only if (p,e) = (2,1). Additional invariants are also determined, including the metric dimension beta = N(m - 1), clique number omega = m, chromatic number chi = m, independence number alpha = N, and domination number gamma_d = N. The theory is illustrated using the family UT3(Fp) of 3 x 3 upper triangular matrices with constant diagonal, yielding d = 3, e = 1, N = p + 1, and m = p^2(p - 1). In particular, G(UT3(F2)) is shown to consist of three disjoint copies of K4. Comparisons with known classifications of rings of orders p^4 and p^5 confirm the validity of the obtained formulas.