Let $\mathcal{L}$ be a lattice. The annihilating-ideal graph of $\mathcal{L}$ is a simple graph whose vertex set is the set of all nontrivial ideals of $\mathcal{L}$ and whose two distinct vertices I and J are adjacent if and only if $I\wedge J=0$. In this paper, crosscap two annihilating-ideal graphs of lattices with at most four atoms are characterized. These characterizations provide the classes of multipartite graphs, which are embedded in the Klein bottle.

In this paper, we examine the interplay between the structural and spectral properties of the unit graph $G\left({\mathbb{Z}}_n\right)$ for $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $p_1,p_2,\dots ,p_k$ are distinct primes and $k,r_1,r_2,\dots ,r_k$ are positive integers such that at least one of the $r_i$ must be greater than 1. We first analyze the structure of the unit graph of ${\mathbb{Z}}_n$, treating it as what we will define as a ‘generalized join graph’ under these conditions. We then determine the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ and deduce that it is integral for all n. Consequently, we obtain the Laplacian spectral radius and algebraic connectivity of $G\left({\mathbb{Z}}_n\right)$. We also prove that the vertex connectivity of $G\left({\mathbb{Z}}_{pq}\right)$ is $(p-2)q$, where $2\ne p<q$. We deduce the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ when $n=p^r{q}^s$, where $2\ne p<q$ are primes and $r,s$ are positive integers. Finally, we present conjectures regarding the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ when $n=p_1{p}_2\dots p_k$ and $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $p_i$ are distinct primes, $r_i$ are positive integers, and $1\le i\le k$.

The purpose of this article is to provide mathematical formulas for some operations on the objects of a non-trivially associated tensor category constructed from a factorization of a group into a subgroup and a set of left coset representatives. A detailed example is provided.

In this paper, we examine the interplay between the structural and spectral properties of the unit graph G(Zn) for n=p1r1p2r2...pkrk, where p1,p2,...,pk are distinct primes and k,r1,r2,...,rk are positive integers such that at least one of the ri must be greater than 1. We first analyze the structure of the unit graph of Zn as a generalized join graph under these conditions. We then determine the Laplacian spectrum of G(Zn) and deduce that it is integral for all n. Consequently, we obtain Laplacian spectral radius and algebraic connectivity of G(Zn). We also prove that the vertex connectivity of G(Zpq) is (p−2)q, where 2≠p<q. We deduce the vertex connectivity of G(Zn) when n=prqs, where 2≠p<q are primes and r,s are positive integers. Finally, we present conjectures about the vertex connectivity of G(Zn) when n=p1p2...pk and n=p1r1p2r2...pkrk, where pi are distinct primes, ri are positive integers, and 1≤i≤k.