Boolean Graphs Research Articles

Reciprocal eigenvalue properties using the zeta and Möbius functions

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra Bn of order 2n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ(P) of complement-closed and convex subposets P of Bn. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V(P) on P. We then write down the adjacency operator A of the graph Γ(P) as the composition of two linear operators on V(P), namely, the operator that multiplies elements of V(P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P, then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ(P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property.

Determinantal properties of Boolean graphs using recursive approach

The aim of this paper is to study the determinant and inverse of the adjacency matrices of weighted and directed versions of Boolean graphs. Our approach is recursive. We describe the adjacency matrix of a weighted Boolean graph in terms of the adjacency matrix of a smaller-sized weighted Boolean graph. This allows us to compute the determinant and inverse of the adjacency matrix of a weighted Boolean graph recursively. In particular, we show that the determinant of a directed Boolean graph is 1. Further, using a classical theorem of Cayley which expresses the determinant of any skew-symmetric matrix as a square of its Pfaffian, we show that for any directed Boolean graph, the characteristic polynomial has all its even degree coefficients strictly positive with the odd ones being zero.

Gene regulatory accelerators on cloud FPGA

SummaryGene regulatory networks (GRN) are dynamic models in time and space. These models are used to predict diseases and in drugs research. GRN models are discrete, and Boolean graphs can efficiently represent them. However, GRN algorithms explore a large solution space with high computational complexity. This work proposes efficient FPGA‐based accelerators to implement two GRN algorithms: attractor computation and Derrida plot. Nevertheless, FPGA accelerator design and deployment are still a challenge. This work presents an accelerator design framework for AWS Amazon FPGA cloud. The framework simplifies the software (SW) and hardware (HW) generation for GRN accelerators. The user provides a high‐level model for the Boolean GRN, and our tool automatically creates AWS‐ready‐to‐deploy software and hardware components. For the attractor and the Derrida plot computation, the proposed FPGA accelerators are on average and faster than a V100 GPU.

ON METRIC DIMENSION OF BOOLEAN GRAPH BG1(G)

Let G be a simple graph with vertex set V and edge set E. BG,NINC,Kq (G), known as boolean graph of G-first kind,simply denoted by BG1(G) is defined as the graph with vertex set V E and two vertices are adjacent if and only if they correspond to adjacent vertices in G or to a vertex and an edge in G such that the edge is not incident with the vertex. In this paper we give a bound for metric dimension of BG1(G) and also find expression for metric dimension of boolean graphs of Complete graphs and Star graphs. Finally, an algorithm for finding the metric dimension of BG1(G) is established.

Boolean factor graph model for biological systems: the yeast cell-cycle network

BackgroundThe desire to understand genomic functions and the behavior of complex gene regulatory networks has recently been a major research focus in systems biology. As a result, a plethora of computational and modeling tools have been proposed to identify and infer interactions among biological entities. Here, we consider the general question of the effect of perturbation on the global dynamical network behavior as well as error propagation in biological networks to incite research pertaining to intervention strategies.ResultsThis paper introduces a computational framework that combines the formulation of Boolean networks and factor graphs to explore the global dynamical features of biological systems. A message-passing algorithm is proposed for this formalism to evolve network states as messages in the graph. In addition, the mathematical formulation allows us to describe the dynamics and behavior of error propagation in gene regulatory networks by conducting a density evolution (DE) analysis. The model is applied to assess the network state progression and the impact of gene deletion in the budding yeast cell cycle. Simulation results show that our model predictions match published experimental data. Also, our findings reveal that the sample yeast cell-cycle network is not only robust but also consistent with real high-throughput expression data. Finally, our DE analysis serves as a tool to find the optimal values of network parameters for resilience against perturbations, especially in the inference of genetic graphs.ConclusionOur computational framework provides a useful graphical model and analytical tools to study biological networks. It can be a powerful tool to predict the consequences of gene deletions before conducting wet bench experiments because it proves to be a quick route to predicting biologically relevant dynamic properties without tunable kinetic parameters.

On some applications related with algebraic structures through different well known graphs

Lot of achievements have been done to comprehend the latest phenomena of our life by discussing the interrelationship of two boughs, graph theory and algebra, of mathematics. In the modern time, it seems to be of great importance to study about eminent, extensive and certifiable applications through well known tools of graph theory. Combinatorial commutative algebra, moderately new, quickly creating numerical tool and foremost tendril of mathematics, has been used exhaustively to solve a list of open problems in algebra under the shadow of graphs. It can be helpful, instead of algebraically, to find the metric dimension of zero-divisor graphs, NP-complete problem, related with finite mathematical structures. In this article we shall discuss some graphs, zero-divisor graphs of finite Boolean rings and balanced bipartite graphs of conjugacy closed loops, and their associated applications.

Fuzzy Metric Dimension of Fuzzy Hypercube Qn and Fuzzy Boolean Graphs

Let G = (V, E, ) be a fuzzy graph. Let M be a subset of V. M is said to be a fuzzy metric basis of G if for every pair of vertices x, y  V ‒ M, there exists a vertex w  M such that d (w, x)  d (w, y). The number of elements in M is said to be fuzzy metric dimension (FMD) of G and is denoted by  (G). The elements in M are called as source vertices. In this paper, we study the fuzzy metric dimension of fuzzy hypercube Qn, fuuzy Boolean Graph BG2 (G) and fuzzy Boolean Graph BG3 (G).

Spectra of Boolean Graphs Over Finite Fields of Characteristic Two

AbstractWith entries of the adjacency matrix of a simple graph being regarded as elements of $\mathbb{F}_{2}$, it is proved that a finite commutative ring $R$ with $1\neq 0$ is a Boolean ring if and only if either $R\in \{\mathbb{F}_{2},\mathbb{F}_{2}\times \mathbb{F}_{2}\}$ or the eigenvalues (in the algebraic closure of $\mathbb{F}_{2}$) corresponding to the zero-divisor graph of $R$ are precisely the elements of $\mathbb{F}_{4}\setminus \{0\}$ . This is achieved by observing a way in which algebraic behavior in a Boolean ring is encoded within Pascal’s triangle so that computations can be carried out by appealing to classical results from number theory.

GRAPHICAL AND ANALYTICAL METHODS FOR PROCESSING “BIG DATA” BASED ON THE ANALYSIS OF THEIR PROPERTIES

This article addresses the subject of creating mathematical methods in order to optimize time and computing resources when processing “big data.” One of the ways of solving this problem is the creation of NoSQL systems, an advantage of which is the flexibility of data models as well as the possibility of horizontal scaling, parallel processing and the speed of obtaining results. From the viewpoint of “big data” analysis, there have been other methods developed such as machine learning, artificial intelligence, distributed processing of streams and events, and visual data research technology.Furthermore, the aim of the research is to develop mathematical methods for processing “big data” based on the system analysis of the data structure properties known as “m-tuples based on ordered sets of arbitrary cardinality (OSAC).”The data structure “m-tuples based on OSAC” is the Boolean, which is ordered by right-side enumeration of the elements of the basis set with cardinality n from the lower boundary of the possible change of the index value for each element of the tuple to the upper one. We formulated certain properties for the data structure investigated. These properties result from rules of logic when forming this structure. We also described mathematical methods based on these properties. Boolean graphs are illustrated with drawings and the outlined vertices of the graph correspond to the declared properties of the given data structure. We derived analytical dependencies to determine these Boolean elements. These Boolean elements do not require the execution of algorithms that implement the particular operations of intersection, union, and membership because the desired result is already determined by these properties.The properties of the data structure in question with regards to m-tuples based on OSAC allow us to determine some interdependencies between m-tuples by their location in the structure. Their location is determined by a pair of indices (j, m) without executing computing algorithms. In this case, the time estimate for obtaining results changes from a cubic O(n3) to linear O(n) dependency.

Boolean graphs are Cohen–Macaulay

ABSTRACTFor each Boolean graph Bn, it is proved that both Bn and its complement are vertex decomposable. It is also proved that Bn is an unmixed graph, thus it is also Cohen–Macaulay.

ON RINGS WHOSE ANNIHILATING-IDEAL GRAPHS ARE BLOW-UPS OF A CLASS OF BOOLEAN GRAPHS

For a finite or an infinite set X, let <TEX>$2^X$</TEX> be the power set of X. A class of simple graph, called strong Boolean graph, is defined on the vertex set <TEX>$2^X{\setminus}\{X,{\emptyset}\}$</TEX>, with M adjacent to N if <TEX>$M{\cap}N={\emptyset}$</TEX>. In this paper, we characterize the annihilating-ideal graphs <TEX>$\mathbb{AG}(R)$</TEX> that are blow-ups of strong Boolean graphs, complemented graphs and preatomic graphs respectively. In particular, for a commutative ring R such that AG(R) has a maximum clique S with <TEX>$3{\leq}{\mid}V(S){\mid}{\leq}{\infty}$</TEX>, we prove that <TEX>$\mathbb{AG}(R)$</TEX> is a blow-up of a strong Boolean graph if and only if it is a complemented graph, if and only if R is a reduced ring. If assume further that R is decomposable, then we prove that <TEX>$\mathbb{AG}(R)$</TEX> is a blow-up of a strong Boolean graph if and only if it is a blow-up of a pre-atomic graph. We also study the clique number and chromatic number of the graph <TEX>$\mathbb{AG}(R)$</TEX>.

Metanetworks of artificially evolved regulatory networks

We study metanetworks arising in genotype and phenotype spaces, in the context of a model population of Boolean graphs evolved under selection for short dynamical attractors. We define the adjacency matrix of a graph as its genotype, which gets mutated in the course of evolution, while its phenotype is its set of dynamical attractors. Metanetworks in the genotype and phenotype spaces are formed, respectively, by genetic proximity and by phenotypic similarity, the latter weighted by the sizes of the basins of attraction of the shared attractors. We find that evolved populations of Boolean graphs form tree-like giant clusters in genotype space, while random populations of Boolean graphs are typically so far removed from each other genetically that they cannot form a metanetwork. In phenotype space, the metanetworks of evolved populations are super robust both under the elimination of weak connections and random removal of nodes.

Complemented graphs and blow-ups of Boolean graphs, with applications to co-maximal ideal graphs

For a set X, let 2X be the power set of X. Let BX be the Boolean graph, which is defined on the vertex set 2X \ {X, ?}, with M adjacent to N if M ? N = ?. In this paper, several purely graph-theoretic characterizations are provided for blow-ups of a finite or an infinite Boolean graph (respectively, a preatomic graph). Then the characterizations are used to study co-maximal ideal graphs that are blow-ups of Boolean graphs (pre-atomic graphs, respectively).

Boolean topological graphs of semigroups: the lack of first-order axiomatization

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class 𝒞 of algebras let G(𝒞)={G(A)∣A∈𝒞}. Assume that 𝒞 is a class of semigroups possessing a nontrivial member with a neutral element and let ℋ be the universal Horn class generated by G(𝒞). We prove that the Boolean core of ℋ, i.e., the topological prevariety generated by finite members of ℋ equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ℋ. We derive analogous results when 𝒞 is a class of monoids or groups with a nontrivial member.

Implements of Graph Blow-Up in Co-Maximal Ideal Graphs

In this article, graph blow-up is used to study the implements of the co-maximal graph and the co-maximal ideal graph 𝒞(R) of a commutative ring R. We show that for a ring R with finitely many maximal ideals, the graph 𝒞(R) is a blow-up of a Boolean graph, and we give more detailed structure of the graphs and 𝒞(R) as blow-ups of Boolean graphs. We determine for what rings R, 𝒞(R) or is a finite graph, respectively. In the end, we get some new results for rings by taking advantage of the graph .

The neighbour-integrity of boolean graphs and its compounds

Communication services have traditionally been characterized by high levels of service dependability. Today cable cuts, node interruptions, software errors or hardware failures and transmission failure at various points can cause interrupted service for long periods of time. Large amounts of data transmitted through fiber optic and satellite communication networks, requires greater degrees of stability or less vulnerability. The neighbour integrity is a vulnerability measure of a graph which is considered as a communications network model. This paper presents neighbour integrity of a Boolean graph. Also, theorems are given for some compounds of Boolean graphs which are obtained by using the graph operations.

Characteristic polynomials and spectra of Boolean graphs

Three theorems are proved by using fundamental concepts concerned with the eigenvectors and the dimension of the space of eigenvectors and by considering that the Boolean graph Bn is a regular graph of nth degree. The results are discussed by applying these theorems to graphs B 1, B 2, B 3 .It is shown that the positive integer nis the greatest eigenvalue of Bn so that multiplicity of n is one and the negative integer — n is the smallest eigenvalue of Bn so that multiplicity of — nis one. Hence, by making a suitable generalization to the spectrums and characteristic polynomials of graphs B 1, B 2, B 3 .general formulas are presented related with the discovery of all spectrums and characteristic polynomials of graphs Bn (n ϵ Z+).

Model checking and boolean graphs

We describe a method for translating a satisfaction problem of the modal μ-calculus into a problem of finding a certain marking of a boolean graph. By giving algorithms to solve the graph problem, we present a global model checking algorithm for a subset of the modal μ-calculus, which has time-complexity O(| A|| T|), where | A| is the size of the assertion and | T| is the size of the model (a labelled transition system). This algorithm is extended to an algorithm for the full modal μ-calculus running in time O(| A| ad | S| ad−1 | T|), where ad is the alternation depth and | S| is the number of states in the transition system, improving on earlier presented algorithms. Moreover, a local algorithm is presented for alternation depth one. This algorithm runs in time O(| A|| T|log(| A|| T|)) and is also an improvement over earlier algorithms.

Coverings of Boolean graphs

Coverings of Boolean graphs

A note on frucht diagrams, boolean graphs and hamilton cycles

We use techniques based on F-diagrams to determine Hamilton cycles in some Boolean graphs.