On Generalized von Neumann Inverse Graphs of Finite Commutative Regular Rings

Let $R$ be a ring with identity.

The generalized von Neumann inverse graph of $R$, denoted by $\Gamma_{Reg}(R)$, is defined as the graph whose vertex set is $Reg(R)$, where two distinct vertices $a,b \in R$ are adjacent if and only if $aba=a$ or $bab=b$.

In this work, we consider the reduced graph $\Gamma'_{Reg}(R)$ obtained by restricting the vertex set to $Reg(R)\setminus{0_R}$, so that $\Gamma_{Reg}(R) \cong K_1 + \Gamma'_{Reg}(R)$, allowing the analysis to focus on its nontrivial structure.

We investigate the structure of $\Gamma'_{Reg}(R)$ for finite commutative von Neumann regular rings and establish several results describing its graph-theoretic properties in relation to the algebraic structure of $R$.

In particular, we derive conditions that characterize connectivity, acyclicity, and planarity, and examine structural features such as vertex degrees, girth, and the existence of pendant vertices, along with their algebraic implications.

We also identify circumstances under which $\Gamma'_{Reg}(R)$ exhibits specific graph classes, including paths, cycles, and wheels, as well as the presence of certain induced subgraphs.

Furthermore, an explicit algorithm is provided to construct $\Gamma'_{Reg}(R)$, and connections with the inclusion ideal graph of $R$ are discussed, offering additional insight into the interplay between ring-theoretic properties and graph structures.