The unit-zero divisor graph of a commutative ring

In this paper, we propose a new graph construction associated with a commutative ring with identity. Let R be such a ring. We define the unit-zero divisor graph of R, denoted by G UZ (R), whose vertex set is R. Two distinct vertices x and y are adjacent if and only if x + y is a unit in R and xy is a zero divisor. This combined adjacency condition incorporates both additive and multiplicative structures of the ring, resulting in a graph that simultaneously reflects these two operations. We investigate several fundamental properties of G UZ (R), including regularity, bipartiteness, planarity, and Hamiltonicity. Moreover, we analyze how these graph-theoretic properties are governed by algebraic components of the ring, such as the group of units, the zero divisor set, ideals, and the Jacobson radical.