The $(n,k)$-arrangement graph $A(n,k)$ is a graph with all the $k$-permutations of an $n$-element set as vertices where two $k$-permutations are adjacent if they agree in exactly $k-1$ positions. We introduce a cyclic decomposition for $k$-permutations and show that this gives rise to a very fine equitable partition of $A(n,k)$. This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of $A(n,k)$. Consequently, we determine the eigenvalues of $A(n,k)$ for small values of $k$. Finally, we show that any eigenvalue of the Johnson graph $J(n,k)$ is an eigenvalue of $A(n,k)$ and that $-k$ is the smallest eigenvalue of $A(n,k)$ with multiplicity ${\cal O}(n^k)$ for fixed $k$.

In the classical Economic Order Quantity (EOQ) model, the common unrealistic assumptions are that all the purchased items are of perfect quality and the demand is constant. However, in a real-world environment, a portion of the purchased items might be damaged due to mishandling or an accident during the shipment process, and the demand rate may increase or decrease over time. Many companies are torn between repairing or replacing the imperfect items with new ones. The right decision on that options is crucial in order to guarantee that there is no shortage of stocks while at the same time not jeopardising the items’ quality and maximising the company’s profit. This paper investigates two EOQ models for imperfect quality items by assuming the demand rate varies with time. Under Policy 1, imperfect items are sent for repairs at an additional cost to the makeup margin; under Policy 2, imperfect items are replaced with equivalent quality items from a local supplier at a higher price. Two mathematical models are developed, and numerical examples along with sensitivity analyses are provided to illustrate these models. Our results reveal that Policy 1 is preferable to Policy 2 most of the time. However, Policy 2 outperforms Policy 1 if there is no minimum threshold on the purchased stock quantity. This research allows a company to discover solutions to previously identified inventory problems and make the inventory-patching process more controlled.

Group signatures are a leading competing signature technique with a substantial amount of research. With group settings, group signatures provide user anonymity. Any group member with access to the group can generate a signature while remaining anonymous. The group manager, however, has the authority to expose and identify the signer if required. Since the privacy of the sender should be preserved, this is a conflict between privacy and accountability. Concerning high performance on security, we propose a novel, well-balanced security and privacy group signature scheme based on a general linear group over group ring. To the best of our knowledge, our work represents the first comprehensive framework for a group signature scheme that utilizes generic linear groups over group rings. We demonstrate that the competing security goals of message trustworthiness, privacy, and accountability are effectively resolved by our protocol. The results of the performance evaluation and simulation demonstrate that our protocol achieves strong security, system robustness, and high-performance efficiency, making it suitable for practical applications.

The lazy cop number is the minimum number of cops needed for the cops to have a winning strategy in the game of Cops and Robbers where at most one cop may move in any round. This variant of the game of Cops and Robbers, called Lazy Cops and Robbers, was introduced by Offner and Ojakian, who provided bounds for the lazy cop number of the hypercube. In this paper, we are interested in the game of Lazy Cops and Robbers on generalized hypercubes. Generalizing existing methods, we will give asymptotic upper and lower bounds on the lazy cop number of the generalized hypercube.