Complement Of Graph Research Articles

Fuzzy equality co-neighborhood domination of graphs

In that paper the fuzzy equality co-neighborhood domination and denoted by $gamma_{en}(G)$ for a new definition of domination was described for the fuzzy graph. This new definition was studied in a strong fuzzy graph and constraints were found for many several graphs. Complementary strong fuzzy graphs of the same graphs were examined and studied in detail.

The F-index and coindex of V-Phenylenic Nanotubes and Nanotorus and their molecular complement graphs

The F-index and coindex of V-Phenylenic Nanotubes and Nanotorus and their molecular complement graphs

From Time–Frequency to Vertex–Frequency and Back

The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples.

Colorings of complements of line graphs

AbstractOur purpose is to show that complements of line graphs (of graphs) enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph that have a chromatic number equal to its chromatic number, namely . In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the topological method in graph theory. We prove the NP‐hardness of deciding the equality between the chromatic number and any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs in this article are elementary and we also provide a short discussion on the ability for the topological methods to cover some of our results.

Generalised online colouring problems in overlap graphs

In this paper we consider an online version of different colouring problems in overlap graphs, motivated by some stacking problems. The instance is a system of time intervals presented in non-decreasing order of the left endpoint. We consider the usual colouring problem as well as b-bounded colouring (colour class have a maximum capacity b) and the same problems in the complement graph. We also consider the case where at most b intervals of the same colour can intersect. For all these versions we obtain a logarithmic competitive ratio w.r.t. the maximum ratio of interval lengths while the best known ratio for the usual colouring was linear. To our knowledge it is the first time the other variants are considered in online overlap graphs. Moreover, in the offline case, a pre-processing allows us to deduce a logarithmic approximation ratio w.r.t. the maximum number of pairwise disjoint intervals in the system. Our method is based on a partition of the overlap graph into permutation graphs, leading to a competitive-preserving reduction of the problem in overlap graphs to the same problem in permutation graphs. We think that this new partition problem by itself is of interest for future work.

Energies of complements of unitary one-matching bi-Cayley graphs over commutative rings

In this paper, we determine the energies of complements of unitary one-matching bi-Cayley graphs over finite commutative rings. We also determine the energies of some one-matching bi-gcd-graphs as well as the energies of complements of some one-matching bi-gcd-graphs, which are generalizations of unitary one-matching bi-Cayley graphs.

Restrained Whole Domination in Graphs

In this paper, we introduce a new definition of domination number in graphs called restrained whole domination number and denoted byɣrwh(G). Some bounds and results for restrained whole domination number are established. Moreover, this number is computed to the complement of certain graphs. Finally, the Effective the deletion or addition an edge or the deletion of a vertex of the graph has this number is discussed.

Minimal obstructions to [formula omitted]-polarity in cographs

A graph is a cograph if it does not contain a 4-vertex path as an induced subgraph. An (s,k)-polar partition of a graph G is a partition (A,B) of its vertex set such that A induces a complete multipartite graph with at most s parts, and B induces the disjoint union of at most k cliques with no other edges. A graph G is said to be (s,k)-polar if it admits an (s,k)-polar partition. The concepts of (s,∞)-, (∞,k)-, and (∞,∞)-polar graphs can be analogously defined.Ekim, Mahadev and de Werra pioneered in the research on polar cographs, obtaining forbidden induced subgraph characterizations for (∞,∞)-polar cographs, as well as for the union of (∞,1)- and (1,∞)-polar cographs. Recently, a recursive procedure for generating the list of cograph minimal (s,1)-polar obstructions for any fixed integer s was found, as well as the complete list of (∞,1)-polar obstructions. In addition to these results, complete lists of minimal (s,k)-polar cograph obstructions are known only for the pair (2,2).In this work we are concerned with the problem of characterizing (∞,k)-polar cographs for a fixed k through a finite family of forbidden induced subgraphs. As our main result, we provide complete lists of forbidden induced subgraphs for the cases k=2 and k=3. Additionally, we provide a partial recursive construction for the general case. By considering graph complements, these results extend to (s,∞)-polar cographs.

Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

We offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least frac{n+1}{n-1} provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size frac{n-1}{2}. With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most frac{n-1}{2}.

Graphs with large diameter and their two distance forcing number

For a given simple graph G = (V,E), the two distance forcing number Z 2d (G) is defined as the minimum cardinality among all Z 2d -sets in G. This paper examine about Z 2d (G) of some graphs with large diameter. Also we determine the 2-distance forcing number of some complement graphs.

On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

Edge ideals with almost maximal finite index and their powers

A graded ideal I in \(\mathbb {K}[x_1,\ldots ,x_n]\), where \(\mathbb {K}\) is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree \(\mathrm {pd}(I)-2\), while it is not linear at the homological degree \(\mathrm {pd}(I)-1\), where \(\mathrm {pd}(I)\) denotes the projective dimension of I. In this paper, we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals the property of having almost maximal finite index does not depend on the characteristic of \(\mathbb {K}\). We also compute the nonlinear Betti numbers of these ideals. Finally, we show that for the edge ideal I of a graph G with almost maximal finite index, the ideal \(I^s\) has a linear resolution for \(s\ge 2\) if and only if the complementary graph \(\bar{G}\) does not contain induced cycles of length 4.

Shannon Capacity and the Categorical Product

Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore $C_{\rm OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds for every pair of graphs, where $F\times G$ is the categorical product of graphs $F$ and $G$. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.

Characterizing dissimilarity of weighted networks

Measuring the dissimilarities between networks is a basic problem and wildly used in many fields. Based on method of the D-measure which is suggested for unweighted networks, we propose a quantitative dissimilarity metric of weighted network (WD-metric). Crucially, we construct a distance probability matrix of weighted network, which can capture the comprehensive information of weighted network. Moreover, we define the complementary graph and alpha centrality of weighted network. Correspondingly, several synthetic and real-world networks are used to verify the effectiveness of the WD-metric. Experimental results show that WD-metric can effectively capture the influence of weight on the network structure and quantitatively measure the dissimilarity of weighted networks. It can also be used as a criterion for backbone extraction algorithms of complex network.

Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs

The reciprocal complementary distance ($RCD$) matrix of a graph $G$ is defined as $RCD(G) = [r_{ij}]$, where $r_{ij} = \frac{1}{1+D-d_{ij}}$ if $i \neq j$ and $r_{ij} = 0$, otherwise, where $D$ is the diameter of $G$ and $d_{ij}$ is the distance between the vertices $v_i$ and $v_j$ in $G$. The $RCD$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of $RCD$-matrix. Two graphs are said to be $RCD$-equienergetic if they have same $RCD$-energy. In this paper, the $RCD$-energy of the complement of line graphs of certain regular graphs in terms of the order and degree is obtained and as a consequence, pairs of $RCD$-equienergetic graphs of same order and having different $RCD$-eigenvalues are constructed.

Anomalous IoT Sensor Data Detection: An Efficient Approach Enabled by Nonlinear Frequency-Domain Graph Analysis

The detection of anomalous Internet-of-Things (IoT) sensor data is extremely important in many industrial applications due to the catastrophic consequences of the faulty or unreliable sensor data. Good anomalous data detection performance with high detection efficiency is indeed a dilemma since it is difficult to derive an explicit detection function to characterize the relationships between the anomalous values and the detection indicator. To overcome this difficulty, the location information of the IoT sensors is exploited in this study to characterize and reconstruct the relationships among the sensor data based on a second-order nonlinear polynomial graph filter (NPGF). The analysis of the sensor data reconstruction model is then conducted in the frequency domain based on the 2-D inverse graph Fourier transform (GFT), and the reconstruction error function for the sensor data is analytically derived based on the second-order GFT coefficients. It is shown that the detection efficiency can be greatly improved if the input graph signal is designed to be bandlimited. The anomalous sensor data detection is then conducted in the frequency domain as high-frequency components are more sensitive to the deviation values. An NPGF-based frequency-domain algorithm is proposed for the anomalous sensor data detection, which is illustrated and validated with a real-world data set for temperature monitoring. The simulation results demonstrate the detection performance and efficiency improvement of the proposed algorithm in anomaly detection.

Partial Phase Cohesiveness in Networks of Networks of Kuramoto Oscillators

Partial, instead of complete, synchronization has been widely observed in various networks, including, in particular, brain networks. Motivated by data from human brain functional networks, in this article, we analytically show that partial synchronization can be induced by strong regional connections in coupled subnetworks of Kuramoto oscillators. To quantify the required strength of regional connections, we first obtain a critical value for the algebraic connectivity of the corresponding subnetwork using the incremental two-norm. We then introduce the concept of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">generalized complement graph</i> , and obtain another condition on the node strength by using the incremental <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\infty$</tex-math></inline-formula> -norm. Under these two conditions, regions of attraction for partial phase cohesiveness are estimated in the forms of the incremental two- and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\infty$</tex-math></inline-formula> -norms, respectively. Our result based on the incremental <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\infty$</tex-math></inline-formula> -norm is the first known criterion that applies to noncomplete graphs. Numerical simulations are performed on a two-level network to illustrate our theoretical results; more importantly, we use real anatomical brain network data to show how our results may contribute to a better understanding of the interplay between anatomical structure and empirical patterns of synchrony.

Bounds for complete cototal domination number of Cartesian product graphs and complement graphs

A total dominating set [Formula: see text] is said to be a complete cototal dominating set if [Formula: see text] has no isolated nodes and it is represented by [Formula: see text]. The complete cototal domination number, represented by [Formula: see text], is the minimum cardinality of a [Formula: see text] set of [Formula: see text]. In this paper, the bounds for complete cototal domination number of Cartesian product graphs and complement graphs are determined.

A unified construction of semiring-homomorphic graph invariants

It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a new characterization of the Shannon capacity \(\Theta \) via Strassen’s Positivstellensatz: \(\Theta ({\bar{G}}) = \inf _f f(G)\), where \(f: \mathsf {Graph}\rightarrow \mathbb {R}_+\) ranges over all monotone semiring homomorphisms. Constructing and classifying graph invariants \(\mathsf {Graph}\rightarrow \mathbb {R}_+\) which are monotone under graph homomorphisms, additive under join, and multiplicative under disjunctive product is therefore of major interest. We call such invariants semiring-homomorphic. The only known such invariants are all of a fractional nature: the fractional chromatic number, the projective rank, the fractional Haemers bounds, as well as the Lovász number (with the latter two evaluated on the complementary graph). Here, we provide a unified construction of these invariants based on linear-like semiring families of graphs. Along the way, we also investigate the additional algebraic structure on the semiring of graphs corresponding to fractionalization. Linear-like semiring families of graphs are a new notion of combinatorial geometry different from matroids which may be of independent interest.

On graphs with no induced five‐vertex path or paraglider

AbstractGiven two graphs and , a graph is ‐free if it contains no induced subgraph isomorphic to or . For a positive integer , is the chordless path on vertices. A paraglider is the graph that consists of a chorless cycle plus a vertex adjacent to three vertices of the . In this paper, we study the structure of (, paraglider)‐free graphs, and show that every such graph satisfies , where and are the chromatic number and clique number of , respectively. Our bound is attained by the complement of the Clebsch graph on 16 vertices. More strongly, we completely characterize all the (, paraglider)‐free graphs that satisfies . We also construct an infinite family of (, paraglider)‐free graphs such that every graph in the family has . This shows that our upper bound is optimal up to an additive constant and that there is no ‐approximation algorithm for the chromatic number of (, paraglider)‐free graphs for any .