Complement Of Graph Research Articles

On the zero forcing number of the complement of graphs with forbidden subgraphs

Zero forcing and maximum nullity are two important graph parameters which have been laboriously studied in order to aid in the resolution of the Inverse Eigenvalue problem. Motivated in part by an observation that the zero forcing number for the complement of a tree on n vertices is either n−3 or n−1 in one exceptional case, we consider the zero forcing number for the complement of more general graphs under certain conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs. Finally, we yield equality between the maximum nullity and zero forcing number of several families of graph complements considered.

Geodetic numbers of tensor product and lexicographic product of graphs

A shortest u - v path between two vertices u and v of a graph G is a u - v geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some u - v geodesic of G. For a nonempty subset S of V ( G ) , let I ( S ) = ∪ u , v ∈ S I [ u , v ] . If I ( S ) = V ( G ) , then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by g ( G ) . In this paper, the exact geodetic numbers of the product graphs T × K m and T ° K ¯ m are obtained, where T is a tree, K ¯ m denotes the complement of the complete graph K m and, × and ° denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.

Enhancing Collaboration in Heterogeneous Multiagent Systems Through Communication Complementary Graph.

Heterogeneous multiagent systems are characterized by diverse task distributions, which are prevalent in practical scenarios, such as distributed decision making and robotic collaboration. A significant challenge in these systems is the constraint of limited observations, where each agent has access only to partial information. Many studies facilitate information exchange by employing shared parameters among agents. However, this approach is generally more effective for homogeneous systems where agents have similar observation or action spaces. In heterogeneous systems, indiscriminate parameter sharing can significantly increase the exploration cost required for effective adaptation. To address this challenge, we propose a novel communication complementary graph model (CCGM) for enhancing collaboration in heterogeneous multiagent systems. Our approach builds upon the training framework of heterogeneous agent reinforcement learning (HARL) with trust region learning and nonparameter sharing. This model utilizes advantage function decomposition and sequential updates to promote policy convergence. Within this framework, we introduce a novel communication method inspired by signaling games, where agents acting as receivers, process messages from other agents alongside their own observations. CCGM aligns the messages with observations in a graph-based communication module, which establishes communication relationships and supplements observational information. Subsequently, agents generate self-interested information, which they then share with others as senders. We evaluate our algorithm across various environments, including multiagent particle environments (MPE) and multiagent MuJoCo (MAMuJoCo) robot experiments. The results demonstrate the effectiveness of CCGM in enhancing HARL-based algorithms.

Synthesis of High-Selectivity Two-Dimensional Filter Banks Using Sigmoidal Function

This paper explores an application of the sigmoidal function—the complementary integral Gaussian error function (erfc(.))—in two-dimensional (2D) uniform and nonuniform filter bank synthesis. The complementary integral Gaussian error function graph represents a smooth low-pass filter magnitude response. A parameter changes the function slope and increases the magnitude response selectivity. The theory is applied to 2D band-pass filter banks. Exact expressions for the magnitude response parameters are determined. As a result, 2D uniform and nonuniform filter banks with very high selectivity and exact shapes are obtained. Three synthesis examples of 2D filter banks with circular and fan-shaped magnitude responses are provided. The theoretical exposition is supplemented with two examples of image analysis using 2D uniform and nonuniform filter banks. A procedure to reduce the computations in image analysis is proposed. A comparison of filter synthesis between Parks–McLellan’s 2D filters and the erfc(.) demonstrates the significantly shorter calculation time of the proposed method.

Exploring the minimum cost conflict mediation path to a desired resolution within the inverse graph model framework

The existing inverse graph model for conflict resolution (GMCR) research primarily concentrates on identifying the required preferences of decisions makers (DMs) such that a desired state is an equilibrium. However, the process of transitioning from the current state to the desired equilibrium is not explored. In this paper, we propose a minimum adjustment cost model taking account of preference adjustment costs to identify the required preferences for a desired state to be an equilibrium. Subsequently, we introduce the concept of transition costs for the first time to quantify expenses involved in guiding a DM transition from one state to another and develop a minimum cost conflict mediation path model. This model aims to identify the most efficient path that minimizes the cost of transitioning from the current state to the desired equilibrium. Moreover, to accommodate the consideration of multiple desired equilibria, we extend the minimum cost conflict mediation path model to analyze and determine the optimal path for transitioning from the current state to one of the identified desired equilibria with the overall minimum cost. Furthermore, to address uncertainty surrounding transition costs, we formulate a probability maximizing conflict mediation path model that considers a limited budget available for the mediation process. Finally, a real-world dispute, the fracking conflict in the province of New Brunswick, Canada, is utilized to demonstrate the application of the proposed models.

Homotopy classification of knotted defects in ordered media

We give a homotopy classification of the global defects in ordered media and explain it via the example of biaxial nematic liquid crystals, that is, systems where the order parameter space is the quotient of the three-sphere S 3 by the quaternion group Q . As our mathematical model, we consider continuous maps from complements of spatial graphs to the space S 3 / Q modulo a certain equivalence relation and find that the equivalence classes are enumerated by the six subgroups of Q . Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of Q ; once we pass to planar diagrams, these labels can be refined to elements of Q associated with each arc. The same classification scheme applies not only in the case of Q but also to arbitrary groups.

Outdegree equitable domination number of certain graph operators

For a vertex u in the dominating set D of a graph G, the number of edges from u to V-D is called the outdegree of u with respect to D, d°D, G(u). A dominating set D is called the outdegree equitable dominating set if the absolute value of the differences of outdegrees of any two vertices in D is at most one. The minimum cardinality of an outdegree equitable dominating set of G is called the outdegree equitable domination number of G, γoe(G). In this paper, we study the outdegree equitable domination number of certain graph operators such as complement, double graph, mycielskian and subdivision of graph.

Metric dimension of the complement of the zero-divisor graph

Metric dimension of the complement of the zero-divisor graph

Efficient computation of Katz centrality for very dense networks via negative parameter Katz

Abstract Katz centrality (and its limiting case, eigenvector centrality) is a frequently used tool to measure the importance of a node in a network, and to rank the nodes accordingly. One reason for its popularity is that Katz centrality can be computed very efficiently when the network is sparse, ie having only O(n) edges between its n nodes. While sparsity is common in practice, in some applications one faces the opposite situation of a very dense network, where only O(n) potential edges are missing with respect to a complete graph. We explain why and how, even for very dense networks, it is possible to efficiently compute the ranking stemming from Katz centrality for unweighted graphs, possibly directed and possibly with loops, by working on the complement graph. Our approach also provides an interpretation, regardless of sparsity, of ‘Katz centrality with negative parameter’ as usual Katz centrality on the complement graph. For weighted graphs, we provide instead an approximation method that is based on removing sufficiently many edges from the network (or from its complement), and we give sufficient conditions for this approximation to provide the correct ranking. We include numerical experiments to illustrate the advantages of the proposed approach.

On rings whose prime ideal sum graphs are line graphs

Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.

Graphs with degree sequence [formula omitted] and [formula omitted]

In this paper we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. We establish various properties of Gm,n and G¯m,n related to recognition, connectivity, diameter, bipartiteness, Hamiltonicity, and pancyclicity. We also show that several classical optimization problems on these graphs are NP-hard, while others can be solved efficiently.

A note on hamiltonicity conditions of the coprime and non-coprime graphs of a finite group

Let $G$ be a group. The coprime and non-coprime graphs of $G$ are introduced by Ma et al. (2014) and Mansoori et al. (2016), respectively, when $G$ is finite. By their definitions, which refer to coprime and non-coprime terms of two positive integers, those graphs must be related. We prove that they are closely related through their graph complement and preserve the isomorphism groups. Furthermore, according to Cayley's theorem, which states that any group $G$ is isomorphic to a subgroup of the symmetric group on $G$, it implies that the studies of the coprime and non-coprime graphs of any group $G$ (especially, when $G$ is finite) can actually be represented by the coprime and non-coprime graphs of any subgroup of the symmetric group on $G$. This encourages us to specifically study the hamiltonicity of both kinds of graphs associated with $G$ when $G$ is isomorphic to the symmetric group on $G$.

The Complexity of Octopus Graph, Friendship Graph, and Snail Graph

Graphs are basic structures that represent objects with nodes and relationships between objects with edges. Trees are one of the parts studied in graph theory along with finding the number of spanning trees of a graph such as octopus graph, friendship graph, and snail graph. The complexity of an octopus graph is strongly dependent on the number and length of tentacles, the complexity of a friendship graph is dependent on the number of triangle cycles, and the complexity of a snail graph is dependent on the number of edges and vertices located in the shell-like part of the snail. To calculate the number of spanning trees (τ(G)) of a graph, various calculations can be used, such as the extension of Kirchhoff's formula. The extension of Kirchhoff's formula uses the determinant of the adjacency matrix and degree matrix of the graph complement of a graph. Therefore, this research applies the extension of Kirchhoff's formula to obtain the complexity of octopus graph, friendship graph, and snail graph. From the analysis, it is obtained that for any n≥2, the number of spanning trees of octopus graph and friendship graph are τ(On )=1/5 √5 [((3+√5)/2)^n-((3-√5)/2)^n ] and τ(Fn )=3^n and the number of spanning trees of snail graph is τ(SIn )=2^(n+2)+3n∙2^(n-1) for n≥1.

Notes on the diameter of the complement of the power graph of a finite group

Notes on the diameter of the complement of the power graph of a finite group

Multi-view subspace clustering based on multi-order neighbor diffusion

Multi-view subspace clustering (MVC) intends to separate out samples via integrating the complementary information from diverse views. In MVC, since the structural information in the graph is crucial to the graph learning, most of the existing algorithms construct the superficial graph from the original data by directly measuring the similarity between the first-order complementary nearest neighbors. However, the information provided by the superficial graph structure would be influenced by contaminated or absent samples. To address this problem, in the proposed method, the higher-order complementary neighbor graphs are exploited to discover the latent structural information between the samples, and fusing the latent structural information across different orders to achieve the MVC. Specifically, the higher-order neighbor graphs under different views are leveraged to estimate the missing samples. Then, to integrate the neighbor graphs of different orders, the multi-order neighbor diffusion fusion is proposed. Nevertheless, the above problem of diffusion fusion is an intractable non-convex issue. Thus, to address it, the multi-order neighbor diffusion fusion is considered as a combination problem of the solution under different order, and the heuristic algorithm is leveraged to solve it. In this way, not only the data representation under different view and also the neighbor structure under different order can be diffused under a joint optimization framework, thus the consistency and integral information among various perspectives and orders can be utilized effectively and simultaneously. Experiments on both incomplete and complete multi-view dataset demonstrate the convincingness of the high-order neighborhood structure based subspace clustering scheme by comparing with the existing approaches.

Probability graph complementation contrastive learning

Graph Neural Network (GNN) has achieved remarkable progress in the field of graph representation learning. The most prominent characteristic, propagating features along the edges, degrades its performance in most heterophilic graphs. Certain researches make attempts to construct KNN graph to improve the graph homophily. However, there is no prior knowledge to choose proper K and they may suffer from the problem of Inconsistent Similarity Distribution (ISD). To accommodate this issue, we propose Probability Graph Complementation Contrastive Learning (PGCCL) which adaptively constructs the complementation graph. We employ Beta Mixture Model (BMM) to distinguish intra-class similarity and inter-class similarity. Based on the posterior probability, we construct Probability Complementation Graphs to form contrastive views. The contrastive learning prompts the model to preserve complementary information for each node from different views. By combining original graph embedding and complementary graph embedding, the final embedding is able to capture rich semantics in the finetuning stage. At last, comprehensive experimental results on 20 datasets including homophilic and heterophilic graphs firmly verify the effectiveness of our algorithm as well as the quality of probability complementation graph compared with other state-of-the-art methods.

On non-bipartite graphs with strong reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If A(G) is nonsingular and X=SA(G)−1S−1 is entrywise nonnegative for some signature matrix S, then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G, denoted by G+. A graph G is said to have the reciprocal eigenvalue property (property(R)) if A(G) is nonsingular, and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T, the following conditions are equivalent: a) T+ is isomorphic to T, b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex).Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph G+ exists and G+ is isomorphic to G. It follows that each graph G in this class has property (SR).

Finding multifaceted communities in multiplex networks

Identifying communities in multilayer networks is crucial for understanding the structural dynamics of complex systems. Traditional community detection algorithms often overlook the presence of overlapping edges within communities, despite the potential significance of such relationships. In this work, we introduce a novel modularity measure designed to uncover communities where nodes share specific multiple facets of connectivity. Our approach leverages a null network, an empirical layer of the multiplex network, not a random network, that can be one of the network layers or a complement graph of that, depending on the objective. By analyzing real-world social networks, we validate the effectiveness of our method in identifying meaningful communities with overlapping edges. The proposed approach offers valuable insights into the structural dynamics of multiplex systems, shedding light on nodes that share similar multifaceted connections.

Graph feature fusion driven by deep autoencoder for advanced hyperspectral image unmixing

In this paper, we propose a pioneering approach for blind Hyperspectral Image (HSI) unmixing named Multi-Features Graph Deep Fusion Learning Networks for HSI Unmixing (MF-GDL). Our method leverages the power of multi-feature graph deep fusion learning networks. MF-GDL integrates spectral HSI graph features with complementary spatial graph features, overcoming the fixed window size limitation observed in prior deep learning approaches in capturing spatial relationships effectively. A key contribution lies in the introduction of multi-view graph learning, a novel aspect in HSI unmixing. By incorporating graph similarity-based spatial information, we address an important gap in the existing literature, marking the first attempt, to our knowledge, to tackle this challenge using a graph deep learning method. Furthermore, we introduce auxiliary information based on Extended Morphological Profiles (EMPs) to enrich spatial context beyond high-level spatial relationships between neighboring pixels. This multi-view fusion learning not only improves accuracy in material abundance estimation but also demonstrates the significance of considering multiple perspectives in HSI unmixing. We validate the efficacy of MF-GDL on four real datasets: Jasper Ridge, Samson, Urban, and Apex showcasing its robust performance and highlighting the substantial contribution of the multi-view aspect in advancing HSI unmixing techniques.

First and second K Banhatti indices of a fuzzy graph applied to decision making

Topological index of a graph is a numerical representation linked to the graph or the molecular structure associated with the graph, shedding light on its topological and structural properties. Different topological indices showcase unique attributes rooted in their degree, spectrum, and distance characteristics. Among these, the first and second K Banhatti indices, introduced by Kulli, involving both vertex and edge degrees, have garnered significant scholarly interest, prompting extensive discussions in graph theory. Within this research paper, a thorough exploration delves into the attributes surrounding the fuzzified first and second K Banhatti indices. This exploration is directed towards several bounds of the index and index of various structures like cycles, complete fuzzy graphs, complete bipartite graphs and stars. Relation between a fuzzy graph and its partial subgraphs, fuzzy graph and its complement graph in connection with the indices is also found. Additionally, the research paper investigates the first and second K Banhatti indices concerning graph operations such as the corona product and coalescence. Additionally, the paper outlines an algorithm designed to accurately compute the first and second K Banhatti indices and proposes an application in decision-making.