Cube addition graph: structure and connectivity
International Journal of Development Research
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 x and y are adjacent whenever x+y is a cube in R. We investigate fundamental graph-theoretic properties including degree, regularity, connectivity, bipartiteness, and Hamiltonian path 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 R and C, connected over Z and Z_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).
Copyright©2026, Nidhi Khandelwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce the addition cube graph over a ring R, whose vertices are the elements of R, and two distinct vertices x and y are adjacent whenever x+y is a cube in R. We investigate fundamental graph-theoretic properties including degree, regularity, connectivity, bipartiteness, and Hamiltonian path 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 R and C, connected over Z and Z_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).
