some results on the sum-annihilating essential submodule graph

Consider a commutative ring r with a non-zero identity 1 ̸= 0, and let m be a non-zero unitary module over r. in this document, our goal is to present the sum-annihilating essential submodule graph ae0 r(m) and its subgraph ae1 r(m) of a module m over a commutative ring r which is described in the following way: the vertex set of graph ae0 r(m) (resp., ae1 r(m)) is the collection of all (resp., non-zero proper) annihilating submodules of m and two separate annihilating submodules n and k are connected anytime n +k is essential in m. we study and investigate the basic properties of graphs aei r(m) (i = 0, 1) and will present some related results. additionally, we explore how the properties of graphs interact with the algebraic structures they represent.