Complement Of Graph Research Articles

Nordhaus-gaddum inequalities for anti fuzzy graph

The objective of this paper is to finds the lower and upper bounds of Nordhaus-Gaddum inequalities of fuzzy chromatic number for anti-fuzzy graph. This paper analyzes the chromatic index of complementary anti fuzzy graphs in some cases. A theorem is proved for anti-fuzzy graph to be k-critical. Examples are provided to derive the vertex coloring of these graphs.

Solving the inverse graph model for conflict resolution using a hybrid metaheuristic algorithm

The paper is concerned with one of the most important questions in conflict analysis: How decision makers can strategically interact in a conflict to reach a specified equilibrium? More specifically, an inverse preference optimization model is formulated to solve the problem of adjusting the preferences of decision makers so as to make a specified state an equilibrium. To this end we define an algorithm which incorporates the hybridization of particle swarm optimization and genetic algorithm. The proposed algorithm adopts a random neighbor strategy for population initialization, the elite reservation and mixed selection operation, and a diversity strategy for population update to improve its efficiency and effectiveness when searching for the required preferences. This approach can help decision makers or third parties to focus their resources on guiding them toward preferences that lead to a specified resolution. Finally, a real-world dispute over exporting bulk water from Eastern Canada is used to demonstrate the applicability and effectiveness of the approach.

ACGNet: Action Complement Graph Network for Weakly-Supervised Temporal Action Localization

Weakly-supervised temporal action localization (WTAL) in untrimmed videos has emerged as a practical but challenging task since only video-level labels are available. Existing approaches typically leverage off-the-shelf segment-level features, which suffer from spatial incompleteness and temporal incoherence, thus limiting their performance. In this paper, we tackle this problem from a new perspective by enhancing segment-level representations with a simple yet effective graph convolutional network, namely action complement graph network (ACGNet). It facilitates the current video segment to perceive spatial-temporal dependencies from others that potentially convey complementary clues, implicitly mitigating the negative effects caused by the two issues above. By this means, the segment-level features are more discriminative and robust to spatial-temporal variations, contributing to higher localization accuracies. More importantly, the proposed ACGNet works as a universal module that can be flexibly plugged into different WTAL frameworks, while maintaining the end-to-end training fashion. Extensive experiments are conducted on the THUMOS'14 and ActivityNet1.2 benchmarks, where the state-of-the-art results clearly demonstrate the superiority of the proposed approach.

On Subgraph Complementation to H-free Graphs

For a class $$\mathcal {G}$$ of graphs, the problem Subgraph Complement to $$\mathcal {G}$$ asks whether one can find a subset S of vertices of the input graph G such that complementing the subgraph induced by S in G results in a graph in $$\mathcal {G}$$ . We investigate the complexity of the problem when $$\mathcal {G}$$ is H-free for H being a complete graph, a star, a path, or a cycle. We obtain the following results: Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time $$2^{o(\mid V(G)\mid )}$$ ), assuming the Exponential Time Hypothesis. We show that the complexity results on a graph class $$\mathcal {G}$$ is also true for the class $$\overline{\mathcal {G}}$$ of the complement graphs of $$\mathcal {G}$$ . Therefore, each of the above results mentioned for the H-free class of graphs is also valid for the $$\overline{H}$$ -free class of graphs. It is noteworthy that our results generalize two main results, namely, Subgraph Complement to triangle-free graphs and Subgraph Complement to d-degenerate graphs, and resolves one open question due to Fomin et al. (Algorithmica, 2020).

Hamming distance between the strings generated by adjacency matrix of a subgraph complementary graph and their sum

Let [Formula: see text] be the adjacency matrix of a graph [Formula: see text]. Let [Formula: see text] denote the row entries of [Formula: see text] corresponding to the vertex [Formula: see text] of [Formula: see text]. The Hamming distance between the strings [Formula: see text] and [Formula: see text] is the number of positions in which [Formula: see text] and [Formula: see text] differ. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of subgraph complement of a graph. We also compute sum of Hamming distances between all pairs of strings generated by the adjacency matrix of [Formula: see text].

On Characterization of Graphs Structures Connected with Some Algebraic Properties

In this paper, we have characterized graph structures connected with some algebraic properties. Also, this paper is actually the concatenation of graph theory and algebra. We have introduced left and right inverse graphs of antiautomorphic inverse property loops. Also, there is a connection between bipartite graphs and mathematical structures, commutator subloop, associator subloop, and associative part, the nucleus of the loop, through edge labeling.

Gracefulness of Two Kinds of Unconnected Graphs with Even Vertices

Two kinds of unconnected double fan graphs with even vertices,[see formula in PDF] and [see formula in PDF] [see formula in PDF] were presented. For natural number [see formula in PDF],[see formula in PDF],the two graphs are all graceful graphs, where [see formula in PDF] are even-vertices path,[see formula in PDF] is odd-vertices path,[see formula in PDF] are the complement of graph [see formula in PDF],[see formula in PDF] is the join graph of [see formula in PDF] and [see formula in PDF].

Powers of cycle graph which are k-self complementary and k-co-self complementary

E. Sampath Kumar and L. Pushpalatha [4] introduced a generalized version of complement of a graph with respect to a given partition of its vertex set. Let G = (V,E) be a graph and P = {V₁, V₂,...,Vk} be a partition of V of order k ≥ 1. The k-complement GPk of G with respect to P is defined as follows: For all Vi and Vj in P, i ≠ j, remove the edges between Vi and Vj , and add the edges which are not in G. Analogues to self complementary graphs, a graph G is k-self complementary (k-s.c.) if GPk ≅ G and is k-co-self complementary (k-co.s.c.) if GPk ≅ Ġ with respect to a partition P of V (G). The mth power of an undirected graph G, denoted by Gm is another graph that has the same set of vertices as that of G, but in which two vertices are adjacent when their distance in G is at most m. In this article, we study powers of cycle graphs which are k-self complementary and k-co-self complementary with respect to a partition P of its vertex set and derive some interesting results. Also, we characterize k-self complementary C2n and the respective partition P of V (C2n). Finally, we prove that none of the C2n is k-co-self complementary for any partition P of V (C2n).

On the Generalized $\vartheta$-Number and Related Problems for Highly Symmetric Graphs

This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound for both the $k$-multichromatic number of a graph and the maximum $k$-colorable subgraph problem. We present various properties of $\vartheta_k(G)$, such as that the sequence $(\vartheta_k(G))_k$ is increasing and bounded from above by the order of the graph $G$. We study $\vartheta_k(G)$ when $G$ is the strong, disjunction, or Cartesian product of two graphs. We provide closed form expressions for the generalized $\vartheta$-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs, and orthogonality graphs. Our paper provides bounds on the product and sum of the $k$-multichromatic number of a graph and its complement graph, as well as lower bounds for the $k$-multichromatic number on several graph classes including the Hamming and Johnson graphs.

Informed Patch Enhanced HyperGCN for skeleton-based action recognition

Human skeleton, as a compact representation of action, has attracted numerous research attentions in recent years. However, skeletal data is too sparse to fully characterize fine-grained human motions, especially for hand/finger motions with subtle local movements. Besides, without containing any information of interacted objects, skeleton is hard to identify human–object interaction actions accurately. Hence, many action recognition approaches that purely rely on skeletal data have met a bottleneck in identifying such kind of actions. In this paper, we propose an Informed Patch Enhanced HyperGraph Convolutional Network that jointly employs human pose skeleton and informed visual patches for multi-modal feature learning. Specifically, we extract five informed visual patches around head, left hand, right hand, left foot and right foot joints as the complementary visual graph vertices. These patches often exhibit many action-related semantic information, like facial expressions, hand gestures, and interacted objects with hands or feet, which can compensate the deficiency of skeletal data. This hybrid scheme can boost the performance while keeping the computation and memory load low since only five extra vertices are appended to the original graph. Evaluation on two widely used large-scale datasets for skeleton-based action recognition demonstrates the effectiveness of the proposed method compared to the state-of-the-art methods. Significant accuracy improvements are reported using X-Sub protocol on NTU RGB+D 120 dataset.

Castelnuovo–Mumford regularity of monomial ideals arising from posets

Let P be a finite poset and suppose I is the Stanley–Reisner ideal of the chain complex of P. In this paper, we prove the inequality for every integer s where denotes the sth symbolic power of I. As a consequence, we deduce that if I(G) is the edge ideal of a graph G such that its complementary graph Gc is bipartite, then the regularity of is either 2s or

On the second neighbourhoods of quasi-strongly regular graphs

A quasi-strongly regular graph of grade p with parameters (n,k,λ;μ1,⋯,μp) is a k-regular graph on n vertices such that any two adjacent vertices have λ common neighbours, any two distinct and non-adjacent vertices have μi common neighbours, and for each μi there exist two distinct and non-adjacent vertices sharing μi common neighbours, where i=1,2,⋯,p. This is a combinatorial generalization of a strongly regular graph. In this paper, we focus on quasi-strongly regular graphs of grade 2. We mainly characterize quasi-strongly regular graphs of grade 2 and diameter 2 with disconnected second neighbourhoods of vertices. Moreover, we obtain similar result for the complement of quasi-strongly regular graphs of grade 2.

δ-Complement of a Graph

Let G(V,X) be a finite and simple graph of order n and size m. The complement of G, denoted by G¯, is the graph obtained by removing the lines of G and adding the lines that are not in G. A graph is self-complementary if and only if it is isomorphic to its complement. In this paper, we define δ-complement and δ′-complement of a graph as follows. For any two points u and v of G with degu=degv remove the lines between u and v in G and add the lines between u and v which are not in G. The graph thus obtained is called δ-complement of G. For any two points u and v of G with degu≠degv remove the lines between u and v in G and add the lines between u and v that are not in G. The graph thus obtained is called δ′-complement of G. The graph G is δ(δ′)-self-complementary if G≅Gδ(G≅Gδ′). The graph G is δ(δ′)-co-self-complementary if Gδ≅G¯(Gδ′≅G¯). This paper presents different properties of δ and δ′-complement of a given graph.

Age of transmission-optimal scheduling for state update of multi-antenna cellular Internet of Things

Timely information updates are critical for real-time monitoring and control applications in the Internet of Things (IoT). In this paper, we consider a multi-antenna cellular IoT for state update where a base station (BS) collects information from randomly distributed IoT nodes through time-varying channel. Specifically, multiple IoT nodes are allowed to transmit their state update simultaneously in a spatial multiplex manner. Inspired by age of information (AoI), we introduce a novel concept of age of transmission (AoT) for the sceneries in which BS cannot obtain the generation time of the packets waiting to be transmitted. The deadline-constrained AoT-optimal scheduling problem is formulated as a restless multi-armed bandit (RMAB) problem. Firstly, we prove the indexability of the scheduling problem and derive the closed-form of the Whittle index. Then, the interference graph and complementary graph are constructed to illustrate the interference between two nodes. The complete subgraphs are detected in the complementary graph to avoid inter-node interference. Next, an AoT-optimal scheduling strategy based on the Whittle index and complete subgraph detection is proposed. Finally, numerous simulations are conducted to verify the performance of the proposed strategy.

Lower Bounds of the Resolvent Estrada Indices for Line Graphs and Complementary Graphs

Lower Bounds of the Resolvent Estrada Indices for Line Graphs and Complementary Graphs

Global triple Roman dominating function

Let k be a positive integer and G=(V,E) a graph. A [k]-Roman dominating function is a function f:V→{0,1,2,…,k+1} such that for every v∈V(G) with f(v)<k, f(AN[v])≥|AN(v)|+k, where AN(v) is the set of neighbors of v assigned a non-zero value under f and AN[v]=AN(v)∪{v}. When k=3, the function f is called a triple Roman dominating function (TRD-function). A global triple Roman dominating function (GTRD-function) on a graph G=(V,E) is a TRD-function for both G and its complement graph G¯. The global triple Roman domination number of a graph G is the minimum weight over all GTRD-functions on G. In this paper, we start the study of the global triple Roman domination number and we obtain various tight bounds for it. Moreover, we prove that the global triple Roman domination problem is NP-complete for bipartite and chordal graphs.

On chromatic number and perfectness of fuzzy graph

Perfect graphs have a significant role in graph theory in terms of structure as well as application. This article embarked a comprehensive study on perfect fuzzy graphs attempted to obtain the fuzzy analog of the famous weak and strong perfect graph theorems following due classification of fuzzy graphs in terms of perfectness. The inter-relationship between chromatic numbers of a fuzzy graph and its complement graph has been studied in detail deriving important theories and results. It has been shown that unlike crisp case, for fuzzy graphs there exists one and only one perfect fuzzy graph theorem satisfying both the conditions of strong and weak perfect graph theorems. A detailed study of the perfectness of different types of fuzzy graphs has been carried out comparing the results with those obtained on crisp graphs. It has been shown through illustrative examples how the perfectness of a fuzzy graph and that of its underlying crisp graph is independent of each other. Finally, the use of perfect fuzzy graph on application grounds has been evaluated by using it in analyzing how a huge percentage of population in a city is unknowingly living under a big threat due exposure to electromagnetic radiation emitted from cell-phone towers.

The A α-spectral radius of dense graphs

A c-cyclic graph is a connected graph with n vertices and n + c−1 edges. In this paper, we consider the problem that: in the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices, which graph has the largest -spectral radius. We shall show that, for , the extremal graph must be a graph with an isolated vertex. This implies that the maximum degree of G is larger, then the -spectral radius of its complement graph is also larger. We also determine the structure of the extremal graph with largest -spectral radius among the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices and maximum degree n−1 for and n−2 for , respectively.

Domination Numbers of Inverse Fuzzy Graphs with Application in Decision-Making Problems

Recently, in Borzooei et al. [Inverse fuzzy graphs with applications, to appear in New Mathematics and Natural Computation], we defined the concept of inverse fuzzy graph as a generalization of graph, which is able to answer some problems that graph theory and fuzzy graph theory cannot explain. Now, in this paper, we define the notions of dominating set, domination number and different kinds of dominating sets like strong, weak, total, independent and inverse dominating sets with their domination numbers in inverse fuzzy graphs. Then, we investigate the relations among different kinds of dominating numbers. Finally, we give an application in decision making for service buildings.

Certain classes of complementary equienergetic graphs

The energy of a graph is the sum of the absolute values of the eigenvalues of a graph‎. ‎Two graphs are said to be equienergetic if they have same energy‎. ‎A graph is said to be complementary equienergetic if it is equienergetic with its complement‎. ‎Recently several complementary equienergetic graphs have been identified‎. ‎In this paper‎, ‎we characterize the cycles‎, ‎paths‎, ‎complete bipartite regular graphs and iterated line graphs of regular graphs‎, ‎which are complementary equienergetic‎. .