Complement Of Graph Research Articles

Divisor graph of the complement of Γ(R)

Let [Formula: see text] be the complement of the zero-divisor graph of a finite commutative ring [Formula: see text]. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that [Formula: see text] is a divisor graph if [Formula: see text] is a local ring. It is shown that when [Formula: see text] is a product of two local rings, then [Formula: see text] is a divisor graph if one of them is an integral domain. Further, if [Formula: see text], then [Formula: see text] is a divisor graph.

Fault Tolerant Control of EPS System with Sensor Fault

Taking into account the sensors faults, modeling of electric power steering (EPS) behavior improves steering reliability. Such task becomes difficult considering the nonlinearity of EPS systems. Therefore, in order to ensure the good behavior stability of the EPS, it is important to validate such established models on a real system. Hence, a fault tolerant control (FTC) design is herein proposed by considering an occurring fault at the torsion bar. A Luenberger observer is used to estimate the torque sensor fault. The FTC is performed to compensate this sensor fault by using the inverse bond graph (BG) modeling. Our simulations reveal the importance of proposed control strategy to improve the EPS system performance but also to reduce the system complexity.

Congruences on graph inverse semigroups

Graph inverse semigroups were defined by Ash and Hall in 1975. They found necessary and sufficient conditions for the semigroups to be congruence free. In this paper we give a classification of congruences on a graph inverse semigroup in terms of the underlying graph. As a consequence, we show that the inverse graph semigroup of a finite graph is congruence Noetherian.

A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the μ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here.

On Brouwer's conjecture for the sum of k largest Laplacian eigenvalues of graphs

Let G be a simple graph with n vertices. For 1≤k≤n, denote by Sk(G) the sum of k largest Laplacian eigenvalues of G. It is conjectured by Brouwer that Sk(G)≤e(G)+(k+12), where e(G) is the number of edges in G. This conjecture has been proved to be true for k=1, k=2, k=n−1, and k=n (for all graphs), and for certain classes of graphs such as trees, unicyclic graphs, and bicyclic graphs (for all k∈{1,2,…,n}). In this paper, we show that if Brouwer's conjecture is true for all graphs when k=p(1≤p≤(n−1)/2), then it is also true for all graphs when k=n−p−1, from which Brouwer's conjecture follows automatically for all graphs when k=n−2 and k=n−3. We also show that for a given graph G and its complement G‾, if Sk(G)≤e(G)+(k+12) holds for all k, then Sk(G‾)≤e(G‾)+(k+12) holds for all k as well, which implies that Brouwer's conjecture holds for the complements of trees, unicyclic graphs, and bicyclic graphs (for all k).

Nordhaus-Gaddum-type result on the second largest signless Laplacian eigenvalue of a graph

ABSTRACT Let G be a simple graph of order n with m edges. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of G is . Let be the signless Laplacian eigenvalues of graph G and also let be the largest positive integer such that . Denote by the complement graph of graph G. If , then we prove that . Moreover, if , then .

Generation of maximal fuzzy cliques of fuzzy permutation graph and applications

Fuzzy permutation graph (FPG) plays a significant role in solving real-life problems where the scope and application of crisp permutation graph get limited due to the fuzziness involved in real situations. In this article an algorithm is designed to find out all maximal fuzzy cliques of a FPG. A similar algorithm is developed at the same time to find out all maximal fuzzy independent sets of the said graph. The inter-relationship between FPG and its two types of complements is presented based on the facts and theories established on fuzzy cliques and fuzzy independent sets. The importance of fuzzy cliques of FPG is discussed through an application on a daily-life problem. The unique property of a FPG having two types of complement graphs help us to solve many useful problems. Here, we have shown how this property of FPG helps us to overcome a hazardous situation occurred due to disrupted scheduling of trains on a foggy day.

On Some closed operations in threshold graphs

This paper is discussing threshold graph. Threshold graph have numerous applications in diverse areas which include computer science and psychology. A threshold graph can be constructed through by repeated application starts with a single vertex, at each step, either a dominating vertex is added, or an isolated vertex is added. The purpose of this paper is to apply some graph operations in a threshold graph and investigating the new graph as result of those operations whether the resulting graph is still a threshold graph or not. We also construct its creation sequence generally.By applying some operations which are given at threshold graph, we can conclude that for union, intersection, ring sum and complement graph, the new graphs are still threshold graphs. Otherwise for the join, the new graphs is not threshold graphs. The creation sequence is achieved generally for each operation.

On the character degree graph of finite groups

Given a finite group $$G$$, let $$\mathrm{cd}(G)$$ denote the set of degrees of the irreducible complex characters of $$G$$. The character degree graph of $$G$$ is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in $$\mathrm{cd}(G)$$, two distinct vertices $$p$$ and $$q$$ being adjacent if and only if $$pq$$ divides some number in $$\mathrm{cd}(G)$$. In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is not bipartite. This extends the analysis of Akhlaghi et al. (Proc Am Math Soc 146:1505–1513, 2018), where the solvable case was treated.

Some results on a spanning subgraph of the complement of the annihilating-ideal graph of a commutative reduced ring

Let [Formula: see text] be a commutative ring with identity which is not an integral domain. Let [Formula: see text] denote the set of all annihilating ideals of [Formula: see text] and let us denote [Formula: see text] by [Formula: see text]. For an ideal [Formula: see text] of [Formula: see text], we denote the annihilator of [Formula: see text] in [Formula: see text] by [Formula: see text]. That is, [Formula: see text]. In this note, for any ring [Formula: see text] with [Formula: see text], we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] are joined by an edge if and only if either [Formula: see text] or [Formula: see text]. Let [Formula: see text] be a reduced ring. The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-theoretic properties of [Formula: see text].

Reformulated Reciprocal Degree Distance and Reciprocal Degree Distance of the Complement of the Mycielskian Graph and Generalized Mycielskian

The reformulated reciprocal degree distance is defined for a connected graph G as R¯t(G)=(1/2)∑u,υ∈VG((dG(u)+dG(υ))/(dG(u,υ)+t)),t≥0, which can be viewed as a weight version of the t-Harary index; that is, H¯t(G)=(1/2)∑u,υ∈VG(1/(dG(u,υ)+t)),t≥0. In this paper, we present the reciprocal degree distance index of the complement of Mycielskian graph and generalize the corresponding results to the generalized Mycielskian graph.

Which multicone graphs are determined by their signless Laplacian spectrum? (The proof of a conjecture)

A multicone graph is defined to be the join of a clique and a regular graph. Let Km and denote a complete graph on m vertices and the complement of a complete graph on n vertices, respectively. In [X. Liu and P. Lu, Signless Laplacian spectral characterization of some joins, Electron. J. Linear Algebra 30.1 (2015)], it have been conjectured that the multicone graphs are determined by their signless Laplacian spectrum. In this short note, we prove this conjecture for n ≠ 3.

Homomorphism and Isomorphism in strong neutrosophic graphs

The concept of strong neutrosophic graph is developed with example and some of their properties are investigated in this paper. Some definitions, homomorphism, isomorphism theorems and propositions in strong neutrosophic graphs are established. Basic operations (like union, intersection etc.,) and complement of strong neutrosophic graphs are also derived here.

线图的补图的Wiener指标

线图的补图的Wiener指标

The Zariski topology graph on scheme

Let [Formula: see text] be a quasi-compact scheme and [Formula: see text]. By [Formula: see text] and [Formula: see text], we denote the set of closed points of [Formula: see text] and the closure of the subset [Formula: see text]. Let [Formula: see text] be a nonempty subset of [Formula: see text]. We define the [Formula: see text]-Zariski topology graph on the scheme [Formula: see text], denoted by [Formula: see text], as an undirected graph whose vertex set is the set [Formula: see text], for two distinct vertices [Formula: see text] and [Formula: see text], there is an arc from [Formula: see text] to [Formula: see text], denoted by [Formula: see text], whenever [Formula: see text]. In this paper, we study the connectivity properties of the graph [Formula: see text], we establish the relationship between the connectivity of the graph [Formula: see text] and the structure of irreducible components of the scheme [Formula: see text]. Also, we characterize when the complement graph of the Zariski topology graph [Formula: see text] is a complete multipartite graph.

Minimal obstructions to [formula omitted]-polarity in cographs

Let s,k be non negative integers. A graph G is (s,k)-polar if its vertex set admits a partition (A,B) such that A induces a complete multipartite graph with at most s parts, and B induces a disjoint union of at most k cliques with no other edges. A notable case is that of (1,1)-polar graphs, which are exactly split graphs. The concepts of (s,∞)-, (∞,k)- and (∞,∞)-polar graphs can be analogously defined. A graph is a cograph if it does not contain P4 as an induced subgraph.Ekim, Mahadev and de Werra first investigated polar cographs, obtaining forbidden induced subgraph characterizations for (∞,∞)-polar cographs, and also for the union of (1,∞)- and (∞,1)-polar cographs. So far, for fixed s and k, complete lists of minimal(s,k)-polar cograph obstructions are known only for the pairs (1,1),(1,2),(2,1) and (2,2).In this paper we find the list of cograph minimal (s,1)-polar obstructions for any fixed positive integer s, as well as the list of cograph minimal (∞,1)-polar obstructions. By considering the complement of such graphs, these results extend to (1,k)- and (1,∞)-polar cographs.

Fast fault-tolerant control of electric power steering systems in the presence of actuator fault

Electric power steering is an advanced steering system that uses an electric motor to improve steering comfort of the car. As a result, the failures in the electric motor can lead to additional fault modes and cause damage of the electric power steering system performance. Hence, to ensure the stability of this latter, the present paper proposes a new method to reconfigure the fault control. A novelty approach of fast fault estimation based on adaptive observer is also proposed. Moreover, to guarantee optimal and fast control, a fault-tolerant control based on inverse bond graph modeling is designed to construct the behavior of the nominal system. The simulation and the experimental results on a real electric power steering system reveal the importance of the control strategy and show that the proposed approach works as intended.

DEKOMPOSISI GRAF MATAHARI

Graph theory is a branch of mathematics that facilitates problem solving. There are a lot of researches which concern on this issue. Various kinds of terms are introduced, one of them is graph decomposition. Graph decomposition is sub graphs collection of non-empty G graph {Hi} until Hi = 〈Ei〉 for non-empty sub graph Ei of E (G), where {Ei} is a partition of E (G). Sub graph Hi in decomposition G do not contain of isolated points. If {Hi} is a decomposition of G, it is denoted by .The discussion of graph decomposition can be developed in graph decomposition through various types. One of the types is decomposition of sun graphs. Sun graph is a graph formed from a circle Cn in which each vertex on a circle graph is given one additional vertex with a degree. So, each vertex in sun graph has 3 degrees, except the edge of cortex which only have 1 degree. The sun graph is the result of corona between two graphs, namely a circular graph with n vertex and complement of a complete graph with 1 number of vertex . The sun graph is denoted by where n is the number of vertex in circle graph. If the vertex naming refers to one vertex (with clockwise rules) and additional vertex naming connected to a circle vertex graph (vi), where the additional vertex has a degree of one, then the rule of naming is and sun graph is partitioned into a sub graph H_i = 〈Ei〉 in the form of K 2 where i ≠ j so that H_i∩H_j = ∅, for i = 1,2,3, ..., n with sub graph If every i + 1, i + 2> n has an implicit + 1 and i + 2 will be expressed as an integers 1,2,3, ..., n (mod n), then the sun graph is 2K 2- decomposition. So, for sun graph n ≥3 is 2K2-decomposition. Keywords: Decomposition, Sun Graph .

On Automorphisms of a Distance-Regular Graph with Intersection Array {69, 56, 10; 1, 14, 60}

Let Γ be a distance-regular graph of diameter 3 with eigenvalues θ0 > θ1 > θ2 > θ3. If θ2 = −1, then the graph Γ3 is strongly regular and the complementary graph \({\bar \Gamma _3}\) is pseudogeometric for pGc3(k, b1/c2). If Γ3 does not contain triangles and the number of its vertices v is less than 800, then Γ has intersection array {69, 56, 10; 1, 14, 60}. In this case Γ3 is a graph with parameters (392, 46, 0, 6) and \({\bar \Gamma _2}\) is a strongly regular graph with parameters (392, 115, 18, 40). Note that the neighborhood of any vertex in a graph with parameters (392, 115, 18, 40) is a strongly regular graph with parameters (115, 18, 1, 3) and its existence is unknown. In this paper, we find possible automorphisms of these strongly regular graphs and automorphisms of a hypothetical distance-regular graph with intersection array {69, 56, 10; 1, 14, 60}. In particular, it is proved that the latter graph is not ar-ctransitive.

The Smith group and the critical group of the Grassmann graph of lines in finite projective space and of its complement

We compute the elementary divisors of the adjacency and Laplacian matrices of the Grassmann graph on $2$-dimensional subspaces in a finite vector space. We also compute the corresponding invariants of the complementary graphs.