This research article focuses on the co-intersection graph of a commutative ring R. The co-intersection graph of R is a simple graph with vertices that are non-trivial ideals of R, and two distinct vertices I and J are adjacent if and only if I + J, R. The objective of the research article is to characterize the Artinian commutative rings R, determining whether their associated co-intersection graph is tree, bipartite, planar, outerplanar, or toroidal. In essence, the article explores the inherent properties of the co-intersection graph and investigates how specific characteristics of Artinian commutative rings influence the resulting graph structures.

In this paper, we characterize trees with a strong secure domination number less than or equal to 4 and compute this parameter for certain classes of graphs. Also, we investigate bounds for the strong secure domination number of vertex gluing of two graphs.

In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ring with nonzero identity, denoted by $\mathrm{\Gamma }\left(R\right)$, is the undirected graph whose vertices are the nonzero zero-divisors of R, and the distinct vertices x and y are adjacent if and only if $xy=0$. Here, we classify the local rings with genus three zero-divisor graphs.

In this series of papers, we study the crosscap two embedding of a class of multipartite graphs, namely, annihilating-ideal graphs of a lattice. In Part 1 of the series [Class of crosscap two graphs arising from lattices-Ⅰ, Mathematics, 11 (2023), 1-26], we classified lattices with the number of atoms less than or equal to 4, whose annihilating-ideal graph can be embedded in the Klein bottle. In this paper, which is Part 2 of the series, we classify all finite lattices with at least 5 atoms whose annihilating-ideal graph is embedded in crosscap two surfaces. These characterizations help us to identify classes of multipartite graphs, which are embedded in the Klein bottle.

Let 𝑛 ≥ 2 be an integer. The graph $\overline{G\left(n\right)}$ is obtained by letting all the elements of {0, … , 𝑛 − 1} to be the vertices and defining distinct vertices 𝑥 and 𝑦 to be adjacent if and only if gcd(𝑥 + 𝑦, 𝑛) ≠ 1. In this paper, we give some bounds for the Castelnuovo–Mumford regularity of the edge ideals and their powers for $\overline{G\left(n\right)}$.