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Let R be a ring. The unit graph of R, denoted by G(R), is the simple graph defined on all elements of R, and where two distinct vertices x and y are linked by an edge if and only if x + y is a unit of R. The diameter of a simple graph G, denoted by diam(G), is the longest distance between all pairs of vertices of the graph G. In the present paper, we prove that for each integer n ≥ 1, there exists a ring R such that n ≤ diam(G(R)) ≤ 2n. We also show that diam(G(R)) ∈ {1, 2, 3, ∞} for a ring R with R/J(R) self-injective and classify all those rings with diam(G(R)) = 1, 2, 3 and ∞, respectively. This extends [12, Theorem 2 and Corollary 1].

For a finite commutative ring R, the square mapping graph of R is a directed graph Γ(R) whose set of vertices is all the elements of R and for which there is a directed edge from a to b if and only if a 2 = b. We establish necessary and sufficient conditions for the existence of isolated fixed points, and the cycles with length greater than 1 in Γ(R). We also examine when the induced subgraph on the set of zero-divisors of a local ring with odd characteristic is semiregular. Moreover, we completely determine the finite commutative rings whose square mapping graphs have exactly two, three or four components.

The peristaltic flow of nanofluids is an emerging area of scientific research due to its vital industrial and medical applications. Herein, a model study is conducted to numerically investigate, the electroosmotically augmented peristaltic pumping of electrically conducting aqueous ionic nanofluid through an inclined asymmetric channel. The behavior of the model for various involved parameters is expressed through a set of graphs. The velocity profile gained a significant increment for Joule heating and electroosmotic velocity parameters. The velocity dropped for the Hartmann number, while a dual behavior is noted for the thermal Grashof number. However, the magnetic number enhances the temperature, while a lower temperature is perceived for the electroosmotic velocity and Debye length parameters. The streamline pattern for the fluid flow subject to multiple values of the electroosmotic velocity parameter can be seen to indicate that for negative values of the parameter, a larger number of streamlines are circulated compared to the case when the electric field is absent, that is, . However, in the case of an opposing electric field, that is, for the positive values of the parameter, the streamline bolus occurs across the centerline of the channel.