Cube Addition Graph: Structure and Connectivity

We introduce the addition cube graph over a ring R, whose vertices are the elements of R, and two distinct vertices xand yare adjacent whenever x+yis a cube in R. We investigate fundamentalgraph-theoreticproperties including degree, regularity, connectivity, bipartiteness, and Hamiltonian paths.For finite fields, we determine conditions under which every element is a cube and analyze the resulting connectivity behavior. In particular, the graph is complete over Rand C, connected over ZandZ_n, and disconnected over F[x]when char⁡(F)=3.We further examine relationships between the graphs of a ring, its ideals, and quotient rings, proving that connectivity of both AC(R/I)and AC(I)implies the connectivity of AC(R).