Dependence Of Number Research Articles

High-speed Source-Device Independent Quantum Random Number Generator on a Chip

High-speed Source-Device Independent Quantum Random Number Generator on a Chip

On the annihilator graphs of partial transformation semigroups

Let X n = { 1 , 2 , … , n } and P n be a partial transformation semigroup on X n . Obviously, the empty set ∅ is a zero element of P n and denoted by 0. Let P n * = P n \\ { 0 } . An element α ∈ P n is a zero divisor of P n if there exists β ∈ P n * such that α β = 0 = β α . The set Ann ( α ) = { β ∈ P n : α β = 0 = β α } is called the annihilator of α in P n . It is clear that 0 ∈ Ann ( α ) for all α ∈ P n . Let Z ( P n ) be the set of all zero divisors of P n and Z ( P n ) * = Z ( P n ) \\ { 0 } . Generally, if α ∈ Z ( P n ) , then there exists β ∈ Z ( P n ) * in which β ∈ Ann ( α ) . In this paper, we construct the annihilator graph Γ n of P n which is an undirected simple graph with vertex set Z ( P n ) * and two distinct vertices α and β are adjacent if and only if Ann ( α ) ∩ Ann ( β ) ≠ { 0 } . Furthermore, we prove some basic structural properties of Γ n and determine invariants for Γ n such as the diameter, girth, clique, domination number, and independence number.

On estimation of extremal entries of the principal eigenvector of a graph

Let X = ( x 1 , x 2 , … , x n ) T be the principal eigenvector corresponding to the spectral radius λ 1 of a graph G of order n. In this paper, we find some bounds on the ratio of the maximal component x max to the minimal component x min of the principal eigenvector X in terms of the graph parameters such as the independence number β , the minimum vertex cover number of the vertex α and the chromatic number χ . Also, we present some bounds on the extremal component x max of the principal eigenvector X. An upper bound of the spectral radius λ 1 of G in terms of the minimum vertex cover number α and order of the graph n is also introduced in this paper.

Machine Learning Predicts Graph Properties, Clique, Girth, and Independent Numbers

Machine Learning Predicts Graph Properties, Clique, Girth, and Independent Numbers

Outer Independent Double Roman Domination Stability in Graphs

An outer independent double Roman dominating function (OIDRDF) on a graph G is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that (i) if f ( v ) = 0 , then the vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f ( w ) = 3 , and if f ( v ) = 1 , then the vertex v must have at least one neighbor w with f ( w ) ≥ 2 and (ii) the subgraph induced by the vertices assigned 0 under f is edgeless. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number γ o i d R ( G ) is the minimum weight of an OIDRDF on G . The γ o i d R -stability ( γ − o i d R -stability, γ + o i d R -stability) of G , denoted by s t γ o i d R ( G ) ( s t − γ o i d R ( G ) , s t + γ o i d R ( G ) ), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the outer independent double Roman domination number. In this paper, we determine the exact values on the γ o i d R -stability of some special classes of graphs, and present some bounds on s t γ o i d R ( G ) . In addition, for a tree T with maximum degree Δ , we show that s t γ o i d R ( T ) = 1 and s t − γ o i d R ( T ) ≤ Δ , and characterize the trees that achieve the upper bound.

Pancyclicity of Hamiltonian graphs

An n -vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from 3 up to n . In 1972, Erdős conjectured that every Hamiltonian graph with independence number at most k and at least n = \Omega(k^{2}) vertices is pancyclic. We prove this old conjecture in a strong form by showing that if such a graph has n = (2+o(1))k^{2} vertices, it is already pancyclic, and this bound is asymptotically best possible.

On the Non-Commuting Graph of the Group U6n

Let G be a finite group. The non-commuting graph of G is a simple graph Γ(G) whose vertices are elements of G∖Z(G), where Z(G) is the center of G, and two distinct vertices a and b are joint by an edge if ab≠ba. In this paper, we study the non-commuting graph of the group U6n. The independent number, clique and chromatic numbers of the non-commuting graph of the group U6n, Γ(U6n), are determined. Additionally, the resolving polynomial, total eccentricity and independent polynomials of Γ(U6n) are computed. Finally, the detour and eccentric connectivity indices of Γ(U6n) are found.

Domination and Independence in Fuzzy Semigraphs

Objectives: To identify dominating and independent sets in fuzzy semigraphs by developing suitable methods. To examine dominating and independent sets in fuzzy semigraphs for their characteristics and behaviors. The goal is to examine the use of the concepts of domination and independence in the fields of analysis of social networks, security of networks, and allocation of resources. Methods: This study introduces the concept of strong arc fuzzy semigraph and explores various parameters such as dominating and independent sets. The Minimum Adjacent Dominating Number (ad-number) \gamma_a(\operatorname{𝓖}) and Maximum Adjacent Dominating Number (ad-number) \Gamma_a(\operatorname{𝓖}) , Minimum End Node Adjacent Dominating Number (ead-number) \gamma_{ea}(\operatorname{𝓖}) and Maximum End Node Adjacent Dominating Number (ead-number) \Gamma_{ae}(\operatorname{𝓖}) as well as Minimum Consecutive Adjacent Dominating Number (cad-number) \gamma_{ca}(\operatorname{𝓖}) and Maximum Consecutive Adjacent Dominating Number (cad-number) \Gamma_{ca}(\operatorname{𝓖}) . These parameters represent the respective minimum and maximum cardinalities taken over all minimal dominating sets of nodes of \operatorname{𝓖} . Additionally, it explores the e-independence dominating number i_e and e-independence number \beta_e(\operatorname{𝓖}) , strongly independence dominating number i_s and strongly independence number \beta_s(\operatorname{𝓖}) and consecutive adjacent independence dominating number i_{ca} and consecutive adjacent independence number \beta_{ca}(\operatorname{𝓖}) . These parameters represent the respective minimum and maximum cardinalities derived from all maximal independent sets of nodes in \operatorname{𝓖} . Findings: The study develops a thorough correlation between fuzzy dominant sets and independent sets in the fuzzy semigraph \operatorname{𝓖} . The statement illustrates the way in which the introduced factors interact and impact the structure and behaviour of the fuzzy semigraph. Novelty: The introduction of strong arc fuzzy semigraphs is a novel notion in the examination of fuzzy graphs. The study involves the creation and examination of novel parameters associated with dominant and independent sets in fuzzy semigraphs. This study aims to establish novel theoretical linkages between fuzzy domination and independent sets, therefore expanding our knowledge of the structural features of fuzzy semigraphs. Keywords: Domination in fuzzy semigraphs, Independent in fuzzy semigraphs, Adjacent-domination, ea-domination, ca-domination, e-independent, s-independent, ca-independent

Degree conditions for path-factors in graphs

A spanning subgraph H of a graph G is called a path-factor if every component of H is a path. Wang and Zhang [RAIRO:RO 57 (2023) 2231–2237] conjectured that a connected graph G with δ(G) ≥ 5 contains a {P2, P5}-factor if δ(G)≥3α(G)−14, where δ(G) and α(G) denote the minimum degree and independence number of G, respectively. We show that the conjecture is true except G ≅ X ∨ 7K3, where X is a spanning subgraph of K3. Furthermore, we give two degree conditions for the existence of {P2, P5}-factors, one of which is a stronger version of Wang’s another conjecture. We also show the degree conditions are best possible.

The domination and independent domination numbers of some families of snarks

The domination and independent domination numbers of some families of snarks

On the <i>k</i>-restricted Intersection Graph

The problem on intersection graph was introduced by Szpilrajn-Marczewski in 1945. This study introduces a new variant of intersection graph, called the $k$-restricted intersection graph. Let $\mathcal{S}_n$ be a nonempty $n$-element set, for some positive integer $n$ and let $S_{(n,k)}$ be the set of all the $k$-element subsets of $\mathcal{S}_n$ where $0\leq k\leq n$. A $k$-restricted intersection graph, denoted by $G_{S_{(n,k)}}$, is a graph with vertex set $S_{(n,k)}$ such that two vertices $A,B\in S_{(n,k)}$ are adjacent whenever $A\cap B\neq \emptyset$ and $A\neq B$. Here, we determined the order and size of $G_{S_{(n,k)}}$. Moreover, some parameters such as independence number, domination number, and isolate domination number of the $k$-restricted intersection graph were established. Finally, necessary and sufficient conditions for a $G_{S_{(n,k)}}$ to be isomorphic to the cycle graph and complete graph were determined.

The inertia bound is far from tight

AbstractThe inertia bound and ratio bound (also known as the Cvetković bound and Hoffman bound) are two fundamental inequalities in spectral graph theory, giving upper bounds on the independence number of a graph in terms of spectral information about a weighted adjacency matrix of . For both inequalities, given a graph , one needs to make a judicious choice of weighted adjacency matrix to obtain as strong a bound as possible. While there is a well‐established theory surrounding the ratio bound, the inertia bound is much more mysterious, and its limits are rather unclear. In fact, only recently did Sinkovic find the first example of a graph for which the inertia bound is not tight (for any weighted adjacency matrix), answering a longstanding question of Godsil. We show that the inertia bound can be extremely far from tight, and in fact can significantly underperform the ratio bound: for example, one of our results is that for infinitely many , there is an ‐vertex graph for which even the unweighted ratio bound can prove , but the inertia bound is always at least . In particular, these results address questions of Rooney, Sinkovic, and Wocjan–Elphick–Abiad.

On the Signless Laplacian ABC-Spectral Properties of a Graph

In the paper, we introduce the signless Laplacian ABC-matrix Q̃(G)=D¯(G)+Ã(G), where D¯(G) is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G. The eigenvalues of the matrix Q̃(G) are the signless Laplacian ABC-eigenvalues of G. We give some basic properties of the matrix Q̃(G), which includes relating independence number and clique number with signless Laplacian ABC-eigenvalues. For bipartite graphs, we show that the signless Laplacian ABC-spectrum and the Laplacian ABC-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian ABC-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian ABC-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q̃(G)−tr(Q̃(G))nI, called the signless Laplacian ABC-energy of G. We obtain some upper and lower bounds for signless Laplacian ABC-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the ABC-energy with the signless Laplacian ABC-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian ABC-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same ABC-energy.

A Proposal for a Mobility-Control Data Transfer Mechanism Based on a Block Network Utilizing End-to-End Authentication Data

Advances in communications technology have had tremendous influence on network implementation to date, and as communication devices get smaller and technologies more intelligent, a great number of solutions are emerging that converge communications with a multitude of domains. In particular, as transportation modes assisting with mobility have developed, many studies are in the pipeline on the domain of mobility. The development of mobility services has prompted the spread of car sharing, which also highlights the need for a system that controls shared vehicles. The data of such control system can be used not only for the operational management of vehicles but also the management of users in terms of user driving analysis or monitoring safe driving. In addition, it can be utilized in a variety of other applications that analyze traffic flow based on vehicle driving data, determine required vehicle maintenance intervals, or schedule vehicle maintenance in advance. However, if forged/altered by a third party, control data might pose a lot of challenges to data utilization. This study proposes a mechanism that manages control data on a blockchain to enhance the integrity of control data created in the communications process. In this paper, we propose a mechanism for applying a blockchain to the communication process between clients and servers. In the proposed mechanism, platform operators generate and provide independent identification information and identification numbers for users. The identification information and identification numbers are used to prove that the generated blocks were created by specific users. Data generated by clients create a block with a hash including user identification information alongside the user identification number. Verification of the generated block is performed on the platform vendor’s server, validating that the user identification number in the block matches the server-side user identification information and the hash of the data within the block. Thus, even when the contents of the block are shared in a public blockchain environment, user identification information is only held by the server and clients, preventing third parties from impersonating specific users to create data blocks. Furthermore, the verification process of the data includes hashes received from other nearby vehicles, preventing data tampering by the block creator.

Dominating induced matchings and other graph parameters

A matching M in a graph G is an induced matching if the largest degree of the subgraph of G induced by M is equal to one. A dominating induced matching (DIM) of G is an induced matching that dominates every edge of G. It is well known that, if they exist, all dominating induced matchings of G are of the same size. The dominating induced matching number of G, denoted by dim ( G ) , is the size of any dominating induced matching of G. In this paper, we continue the study of dominating induced matchings. We prove that, if G has a DIM, then the induced matching number of G is equal to the independence number of its line graph L(G) and to the edge domination number of G. It is also shown that dim ( G ) ≤ 2 dim ( L ( G ) ) , provided that both G and L(G) have a DIM. We also present some bounds on dim ( G ) . In particular, for a tree T with a DIM we show that ⌈ n − l + 1 3 ⌉ ≤ dim ( T ) ≤ ⌊ n − 1 + l 3 ⌋ , where l is the number of leaves. Moreover, for a regular graph G we establish some Nordhaus-Gaddum type bounds.

Design of evolutionary computational intelligent solver for nonlinear corneal shape model by Mexican Hat and Gaussian wavelet neural networks

In this study, an integrated computational intelligence algorithm is implemented for the numerical treatment of the two-point boundary value problems that arise in the nonlinear corneal shape (NCS) model through the exploitation of wavelet neural networks including Mexican-Hat (MHWNNs) and Gaussian-wavelet (GWNNs) through global genetic algorithms (GAs) then hybridization with local sequential quadratic programming (SQP) solvers, i.e. MHWNNs-GAs, GWNNs-GAs, MHWNNs-GA-SQP, and GWNNs-GA-SQP respectively. The GWNNs and MHWNNs are applied to calculate the mean squared error of mathematical modeling of the proposed problem through objective functions while optimization of the fitness functions is initially conducted with an efficiency of global search GAs and then the efficacy of local search technique SQP for fine-tuning. A comparison of the proposed solutions of MHWNNs-GAs, GWNNs-GAs, MHWNNs-GA-SQP, and GWNNs-GA-SQP solvers with a reference solution of Adam’s method shows that the proposed schemes have better accuracy, stability, efficiency consistency on an independent number of runs analyzed through complexity analysis and different statistical operators.

Misiurewicz polynomials and dynamical units, part II

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The maximum number of maximum generalized 4‐independent sets in trees

AbstractA generalized ‐independent set is a set of vertices such that the induced subgraph contains no trees with ‐vertices, and the generalized ‐independence number is the cardinality of a maximum ‐independent set in . Zito proved that the maximum number of maximum generalized 2‐independent sets in a tree of order is if is odd, and if is even. Tu et al. showed that the maximum number of maximum generalized 3‐independent sets in a tree of order is if , and if , and if and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized ‐independent sets in a tree for a general integer . As applications, we show that the maximum number of generalized 4‐independent sets in a tree of order is and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4‐independent sets.

THE NON-BRAID GRAPH OF DIHEDRAL GROUP Dn

THE NON-BRAID GRAPH OF DIHEDRAL GROUP Dn

On conflict-free cuts: Algorithms and complexity

One way to define the Matching Cut problem is: Given a graph G, is there an edge-cut M of G such that M is an independent set in the line graph of G? We propose the more general Conflict-Free Cut problem: Together with the graph G, we are given a so-called conflict graph Gˆ on the edges of G, and we ask for an edge-cutset M of G that is independent in Gˆ. Since conflict-free settings are popular generalizations of classical optimization problems and Conflict-Free Cut was not considered in the literature so far, we start the study of the problem. We show that the problem is NP-complete even when the maximum degree of G is 5 and Gˆ is 1-regular. The same reduction implies an exponential lower bound on the solvability based on the Exponential Time Hypothesis. We also give parameterized complexity results: We show that the problem is fixed-parameter tractable with the vertex cover number of G as a parameter, and we show W[1]-hardness even when G has a feedback vertex set of size one, and the clique cover number of Gˆ is the parameter. Since the clique cover number of Gˆ is an upper bound on the independence number of Gˆ and thus the solution size, this implies W[1]-hardness when parameterized by the cut size. We list polynomial-time solvable cases and interesting open problems. At last, we draw a connection to a symmetric variant of SAT.