Search Results (8)

The study of algebraic invariants associated with Brauer configuration algebras induced by appropriate multisets is said to be a Brauer analysis of the data that define the multisets. In general, giving an explicit description of such invariants as the dimension of the algebras or the dimension of their centers is a hard problem. This paper performs a Brauer analysis on some generators of Thompson’s group F. It proves that such generators and some appropriate Christoffel words induce Brauer configuration algebras whose dimensions are given by the number of edges and vertices of the binary trees defining them. The Brauer analysis includes studying the covering graph induced by a corresponding quiver; this paper proves that these graphs allow for finding set-theoretical solutions of the Yang–Baxter equation. Full article

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. Full article

Brauer configuration algebras are path algebras induced by appropriated multiset systems. Since their structures underlie combinatorial data, the general description of some of their algebraic invariants (e.g., their dimensions or the dimensions of their centers) is a hard problem. Integer partitions and compositions of a given integer number are examples of multiset systems which can be used to define Brauer configuration algebras. This paper gives formulas for the dimensions of Brauer configuration algebras (and their centers) induced by some integer partitions. As an application of these results, we give examples of Brauer configurations, which can be realized as branch data of suitable branched coverings over different surfaces. Full article

This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring $R$ and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set $Z\left(R_E\right)\backslash \left\{\right[0\left]\right\}=R_E\backslash \left\{\right[0],[1\left]\right\}$, where $R_E=\left\{\right[x] :x\in R\}$ and $\left[x\right]=\{y\in R : ann(x)=ann(y\left)\right\}$ is called a compressed zero-divisor graph, denoted by ${\Gamma }_E\left(R\right)$. An edge is formed between two vertices $\left[x\right]$ and $\left[y\right]$ of $Z\left(R_E\right)$ if and only if $\left[x\right]\left[y\right]=\left[xy\right]=\left[0\right],$ that is, iff $xy=0$. For a ring $R$, graph $G$ is said to be realizable as ${\Gamma }_E\left(R\right)$ if $G$ is isomorphic to ${\Gamma }_E\left(R\right)$. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance. Full article

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. Full article

Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, $B\subseteq V\left(\mathsf{\Gamma }\right)$. If the representation of two different edges is unequal, then B is an edge-multiset resolving a set of $\mathsf{\Gamma }$. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of $\mathsf{\Gamma }$. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions. Full article

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. Full article

Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) $T_1,T_2,\dots ,T_n$. In this case, for $1\le j\le n-1$, two consecutive tiles $T_j$ and $T_{j+1}$ share exactly one edge, either the edge at the east (west) of $T_j$ ($T_{j+1}$) or the edge at the north (south) of $T_j$ ($T_{j+1}$). Finding the number of perfect matchings associated with a given snake graph is one of the most remarkable problems regarding these graphs. It is worth noting that such a number of perfect matchings allows a bijection between the set of snake graphs and the positive continued fractions. Furthermore, perfect matchings of snake graphs have also been used to find closed formulas for cluster variables of some cluster algebras and solutions of the Markov equation, which is a well-known Diophantine equation. Recent results prove that snake graphs give rise to some string modules over some path algebras, connecting snake graph research with the theory of representation of algebras. This paper uses this interaction to define Brauer configuration algebras induced by schemes associated with some multisets called polygons. Such schemes are named Brauer configurations. In this work, polygons are given by some admissible words, which, after appropriate transformations, permit us to define sets of binary trees called groves. Admissible words generate codes whose energy values are given by snake graphs. Such energy values can be estimated by using Catalan numbers. We include in this paper Python routines to compute admissible words (i.e., codewords), energy values of the generated codes, Catalan numbers and dimensions of the obtained Brauer configuration algebras. Full article