Complement Of Graph Research Articles

Vertex Degrees and Isomorphic Properties in Complement of an m-Polar Fuzzy Graph

Computational intelligence and computer science rely on graph theory to solve combinatorial problems. Normal product and tensor product of an m-polar fuzzy graph have been introduced in this article. Degrees of vertices in various product graphs, like Cartesian product, composition, tensor product, and normal product, have been computed. Complement and μ-complement of an m-polar fuzzy graph are defined and some properties are studied. An application of an m-polar fuzzy graph is also presented in this article.

On the Complement of the Intersection Graph of Zero-Divisors of the ring Zn

On the Complement of the Intersection Graph of Zero-Divisors of the ring Z_n

A statistical analysis of UK financial networks

In recent years, with a growing interest in big or large datasets, there has been a rise in the application of large graphs and networks to financial big data. Much of this research has focused on the construction and analysis of the network structure of stock markets, based on the relationships between stock prices. Motivated by Boginski et al. (2005), who studied the characteristics of a network structure of the US stock market, we construct network graphs of the UK stock market using same method. We fit four distributions to the degree density of the vertices from these graphs, the Pareto I, Fréchet, lognormal, and generalised Pareto distributions, and assess the goodness of fit. Our results show that the degree density of the complements of the market graphs, constructed using a negative threshold value close to zero, can be fitted well with the Fréchet and lognormal distributions.

Anti-Path Cover on Sparse Graph Classes

We show that it is possible to use Bondy-Chvatal closure to design an FPT algorithm that decides whether or not it is possible to cover vertices of an input graph by at most k vertex disjoint paths in the complement of the input graph. More precisely, we show that if a graph has tree-width at most w and its complement is closed under Bondy-Chvatal closure, then it is possible to bound neighborhood diversity of the complement by a function of w only. A simpler proof where tree-depth is used instead of tree-width is also presented.

Local coloring of self complementary graphs

Let G=(V,E) be a graph. A local coloring of a graph G of order at least 2 is a function c:V(G)⟶N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥ms, where ms is the size of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χℓ(c). The local chromatic number of G is χℓ(G)=min{χℓ(c)}, where the minimum is taken over all local colorings c of G. In this paper we study the local coloring for some self complementary graphs. Also we present a sc-graph with local chromatic number k for any given integer k≥6.

Some New Concepts of Fuzzy Soft Graphs

In this paper, we introduce some new concepts of fuzzy soft graphs with the notions of complement and μ-complement fuzzy soft graphs introduced. Investigating some of their properties, we show that the complement of strong fuzzy soft graph is strong fuzzy soft one as well as the complement of a complete fuzzy soft graph is complete fuzzy soft one. Finally, we state and prove some results related to these concepts.

Inverting Matrix Using a Graph in the Theory of Dynamic Systems

Abstract To obtain the transfer equation in dynamic system theory, we need to calculate an inverse matrix. In this work we show both the classic methods and the methods used in automation theory of inverse matrix calculation. However, they are both time consuming and difficult. In the second section, we will show a simpler method. We will define how to create a graph for a given matrix and how to invert it. The matrix for the inverse graph is inverse of the original. We illustrate this method on an example and compare it to a classic solution.

Reciprocal complementary distance equienergetic graphs

The reciprocal complementary distance (RCD) matrix of a graph [Formula: see text] is defined as [Formula: see text], in which [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the diameter of [Formula: see text] and [Formula: see text] is the distance between the [Formula: see text]th and [Formula: see text]th vertex of [Formula: see text]. The [Formula: see text]-energy [[Formula: see text]] of [Formula: see text] is defined as the sum of the absolute values of the eigenvalues of RCD-matrix of [Formula: see text]. Two graphs [Formula: see text] and [Formula: see text] are said to be RCD-equienergetic if [Formula: see text]. In this paper, we obtain the RCD-eigenvalues and RCD-energy of the join of certain regular graphs and thus construct the non-RCD-cospectral, RCD-equienergetic graphs on [Formula: see text] vertices, for all [Formula: see text].

Complementary Eccentric Uniform Labeling Graphs

Given a graph \(G=(V,E)\), a set \(M\subset V\) is called Complementary Eccentric Uniform (CEU), if the \(M\)-eccentricity labeling \(e_M(u)=\max\{d(u,v):v\in M\}\) is identical for all \(u\in V-M\). The least cardinality of a CEU set is called the CEU number of the graph \(G\). In this paper we initiate a study on CEU labelled graphs and obtain bounds for certain graphs.

Domination in total graphs of small rings

Let [Formula: see text] be a commutative ring and [Formula: see text] be its set of zero divisors. The total graph of [Formula: see text] (introduced by Anderson and Badawi) denoted by [Formula: see text]. For any positive integers [Formula: see text] and [Formula: see text] we obtain the domination number of the total graph of [Formula: see text]. We also obtain various domination parameters including [Formula: see text] and [Formula: see text]. Finally we explore domination parameters in the complement of the total graph [Formula: see text].

Coloring of character graphs

ABSTRACTIn this paper, we study the character graph Δ(G) of a finite solvable group G. We prove that sum of the chromatic number of Δ(G) and the matching number of complement graph of Δ(G) is equal to the order of Δ(G). Also, we prove that when Δ(G) is not a block, the chromatic number of Δ(G) is equal to the clique number of Δ(G).

Colouring perfect graphs with bounded clique number

A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Grötschel, Lovász, and Schrijver [9] from 1988 finds an optimal colouring of a perfect graph in polynomial time. But this algorithm uses the ellipsoid method, and it is a well-known open question to construct a “combinatorial” polynomial-time algorithm that yields an optimal colouring of a perfect graph.A skew partition in G is a partition (A,B) of V(G) such that G[A] is not connected and G‾[B] is not connected, where G‾ denotes the complement graph; and it is balanced if an additional parity condition on certain paths in G and G‾ is satisfied.In this paper we first give a polynomial-time algorithm that, with input a perfect graph, outputs a balanced skew partition if there is one. Then we use this to obtain a combinatorial algorithm that finds an optimal colouring of a perfect graph with clique number k, in time that is polynomial for fixed k.

Reformulated Reciprocal Degree Distance of Transformation Graph

The reformulated reciprocal degree distance is defined for a connected graph G as Rt‾(G)=12∑u,v∈V(G)(d(u)+d(v))dG(u,v)+t,t≥0. The reformulated reciprocal degree distance is a weight version of the t-Harary index, that is, Ht‾(G)=12∑u,v∈V(G)1dG(u,v)+t,t≥0. In this paper, the reformulated reciprocal degree distance and reciprocal degree distance of the Mycielskian graph and the complement of the Mycielskian graph are obtained.

Intuitionistic fuzzy soft graphs with applications

We present different types of new concepts, including intuitionistic fuzzy soft graphs, complete intuitionistic fuzzy soft graph, strong intuitionistic fuzzy soft graph and self complement of intuitionistic fuzzy soft graph. We describe various methods of their construction, and investigate some of their related properties. We also discuss applications of intuitionistic fuzzy soft graphs in communication network and decision making.

A study of industrial electricity consumption based on partial Granger causality network

Abstract The paper studies the industrial energy transferring paths among the industries of China by distinguishing direct causality from the indirect. With complementary graphs, we propose that industrial causal relationship can be heterogeneous, and provide insights for refining robust industrial causality framework. First, by analyzing the in-weight and out-weight of the industries in Granger causality networks we find that public utilities have significant causality with other industries, and the industries with higher degree value tend to have stronger causality with others. Further, we eliminate the exogenous links by partial Granger causality model and find both Granger and partial Granger networks have consistent hub industries while some outliers emerge in partial Granger causality networks. Besides, compared with GX, GZ, HN and YN, the correlation between the volume of electricity consumption and the weight of each industry is more significant in the networks of GD and NF. By studying the characteristics of complementary graphs, we show that the industrial energy transferring paths in GD are more multidimensional, and the corresponding interdependent relationship among industries is more robust. Finally, using bootstrap method we verify the reliability of each industrial relationship network. Results exhibit that GD, GX and NF have more reliable causal relationship networks than other provinces, revealing their industrial structure to be more stable.

Enumeration of unlabeled graphs such that both the graph and its complement are 2-connected

It is well-known that the complement of a disconnected graph is connected, that is, the number of disconnected unlabeled graphs whose complement is also disconnected is zero. By this fact, we can easily express the number of connected unlabeled graphs whose complement is also connected, by the numbers of graphs and connected graphs. The generating functions of them are obtained by Harary [5]. In this paper, we express the number of unlabeled graphs such that both the graph and its complement are 2-connected, by the numbers of graphs whose generating functions are known.

Chromatic Number of Resultant of Fuzzy Graphs

Fuzzy graph coloring techniques are used to solve many complex real world problems. The chromatic number of complement of fuzzy graph is obtained and compared with the chromatic number of the corresponding fuzzy graph. The chromatic number of the resultant fuzzy graphs is studied, obtained by various operations on fuzzy graphs like union, join and types of products. A solution to routing problem is suggested by using chromaticity of fuzzy graphs.

Proper Connection Numbers of Complementary Graphs

A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by $$\textit{pc}(G)$$ , is the minimum number of colors that are needed to make G proper connected. In this paper, we investigate the proper connection number of the complement of a graph G according to some constraints of G itself. Also, we characterize the graphs on n vertices that have proper connection number $$n-2$$ . Using this result, we give a Nordhaus–Gaddum-type theorem for the proper connection number. We prove that if G and $$\overline{G}$$ are both connected, then $$4\le \textit{pc}(G)+\textit{pc}(\overline{G})\le n$$ , and the upper bound holds if and only if G or $$\overline{G}$$ is the n-vertex tree with maximum degree $$n-2$$ .

On Invariant Graph Subspaces

In this paper we discuss the problem of decomposition for unbounded $2\times 2$ operator matrices by a pair of complementary invariant graph subspaces. Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrices.

A Generalization of Join and an Algorithmic Recognition Problem

We consider a new graph operation c-join which generalizes join and co-join. We show that odd hole-free graphs ( odd antihole-free graphs) are closed under c-join and describe a polynomial time algorithm to recognize graphs that admit a c-join. The time complexity of the (a) recognition problem, (b) maximum weight independent set (MWIS) problem, and (c) minimum coloring (MC) problem for odd hole-free graphs are still unknown. Let H be an odd hole-free graph that contains an odd antihole as an induced subgraph and GH be the class of all graphs generated from the induced subgraphs of H by using c-join recursively. Then GH is odd hole-free, contains all P4-free graphs, complement of all bipartite graphs, and some imperfect graphs. We show that the MWIS problem, maximum weight clique (MWC) problem, MC problem, and minimum clique cover (MCC) problem can be solved efficiently for GH .