Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages.

Since its introduction, researching malware has had two main goals. On the one hand, malware writers have been focused on developing software that can cause more damage to a targeted host for as long as possible. On the other hand, malware analysts have as one of their main purposes the development of tools such as malware detection systems (MDS) or network intrusion detection systems (NIDS) to prevent and detect possible threats to the informatic systems. Obfuscation techniques, such as the encryption of the virus’s code lines, have been developed to avoid their detection. In contrast, shallow machine learning and deep learning algorithms have recently been introduced to detect them. This paper is devoted to some theoretical implications derived from these investigations. We prove that hidden algebraic structures as equipped posets and their categories of representations are behind the research of some infections. Properties of these categories are given to provide a better understanding of different infection techniques.

The open shop scheduling problem (OSSP) is one of the standard scheduling problems. It consists of scheduling jobs associated with a finite set of tasks developed by different machines. In this case, each machine processes at most one operation at a time, and the job processing order on the machines does not matter. The goal is to determine the completion times of the operations processed on the machines to minimize the largest job completion time, called C_max. This paper proves that each OSSP has associated a path algebra called Brauer configuration algebra whose representation theory (particularly its dimension and the dimension of its center) can be given using the corresponding C_max value. It has also been proved that the dimension of the centers of Brauer configuration algebras associated with OSSPs with minimal C_max are congruent modulo the number of machines.

Cayley hash values are defined by paths of some oriented graphs (quivers) called Cayley graphs, whose vertices and arrows are given by elements of a group $\mathfrak{H}$. On the other hand, Brauer messages are obtained by concatenating words associated with multisets constituting some configurations called Brauer configurations. These configurations define some oriented graphs named Brauer quivers which induce a particular class of bound quiver algebras named Brauer configuration algebras. Elements of multisets in Brauer configurations can be seen as letters of the Brauer messages. This paper proves that each point $(x,y)\in \mathcal{V}=\mathbb{R}\backslash \left\{0\right\}\times \mathbb{R}\backslash \left\{0\right\}$ has an associated Brauer configuration algebra ${\Lambda }_{{\mathfrak{B}}^{(x,y)}}$ induced by a Brauer configuration ${\mathfrak{B}}^{(x,y)}$. Additionally, the Brauer configuration algebras associated with points in a subset of the form $(\lfloor (x)\rfloor ,\lceil (x)\rceil ]\times (\lfloor (y)\rfloor ,\lceil (y)\rceil ]\subset \mathcal{V}$ have the same dimension. We give an analysis of Cayley hash values associated with Brauer messages $\mathfrak{M}\left({\mathfrak{B}}^{(x,y)}\right)$ defined by a semigroup generated by some appropriated matrices $A_0,A_1,A_2\in \mathrm{GL}(2,\mathcal{R})$ over a commutative ring $\mathcal{R}$. As an application, we use Brauer messages $\mathfrak{M}\left({\mathfrak{B}}^{(x,y)}\right)$ to construct explicit solutions for systems of linear and nonlinear differential equations of the form $X^{″}\left(t\right)+MX\left(t\right)=0$ and $X^{\prime }\left(t\right)-X^2\left(t\right)N\left(t\right)=N\left(t\right)$ for some suitable square matrices, M and $N\left(t\right)$. Python routines to compute Cayley hash values of Brauer messages are also included.

This paper is focused on some algebraic and combinatorial properties of a TMTO (Time–Memory Trade-Off) for a chosen plaintext attack against a cryptosystem with a perfect secrecy property. TMTO attacks aim to retrieve the preimage of a given one-way function more efficiently than an exhaustive search and with less memory than a dictionary attack. TMTOs for chosen plaintext attacks against cryptosystems with a perfect secrecy property are associated with some directed graphs, which can be defined by suitable collections of multisets called Brauer configurations. Such configurations induce so-called Brauer configuration algebras, the algebraic and combinatorial invariant analysis of which is said to be a Brauer analysis. In this line, this paper proposes formulas for dimensions of Brauer configuration algebras (and their centers) induced by directed graphs defined by TMTO attacks. These results are used to provide some set-theoretical solutions for the Yang–Baxter equation.

Seaweeds or seaweed Lie algebras are subalgebras of the full-matrix algebra $\mathrm{Mat}\left(n\right)$ introduced by Dergachev and Kirillov to give an example of algebras for which it is possible to compute the Dixmier index via combinatorial methods. It is worth noting that finding such an index for general Lie algebras is a cumbersome problem. On the other hand, Brauer configuration algebras are multiserial and symmetric algebras whose representation theory can be described using combinatorial data. It is worth pointing out that the set of integer partitions and compositions of a fixed positive integer give rise to Brauer configuration algebras. However, giving a closed formula for the dimension of these kinds of algebras or their centers for all positive integer is also a tricky problem. This paper gives formulas for the dimension of Brauer configuration algebras (and their centers) induced by some restricted compositions. It is also proven that some of these algebras allow defining seaweeds of Dixmier index one.