Graph Operations Research Articles

3D shape knowledge graph for cross‐domain 3D shape retrieval

AbstractThe surge in 3D modelling has led to a pronounced research emphasis on the field of 3D shape retrieval. Numerous contemporary approaches have been put forth to tackle this intricate challenge. Nevertheless, effectively addressing the intricacies of cross‐modal 3D shape retrieval remains a formidable undertaking, owing to inherent modality‐based disparities. The authors present an innovative notion—termed “geometric words”—which functions as elemental constituents for representing entities through combinations. To establish the knowledge graph, the authors employ geometric words as nodes, connecting them via shape categories and geometry attributes. Subsequently, a unique graph embedding method for knowledge acquisition is devised. Finally, an effective similarity measure is introduced for retrieval purposes. Importantly, each 3D or 2D entity can anchor its geometric terms within the knowledge graph, thereby serving as a link between cross‐domain data. As a result, the authors’ approach facilitates multiple cross‐domain 3D shape retrieval tasks. The authors evaluate the proposed method's performance on the ModelNet40 and ShapeNetCore55 datasets, encompassing scenarios related to 3D shape retrieval and cross‐domain retrieval. Furthermore, the authors employ the established cross‐modal dataset (MI3DOR) to assess cross‐modal 3D shape retrieval. The resulting experimental outcomes, in conjunction with comparisons against state‐of‐the‐art techniques, clearly highlight the superiority of our approach.

Equi Neighbor Polynomial of Some Binary Graph Operations

Let \(G(V,E)\) be a simple graph of order \(n\) with vertex set \(V\) and edge set \(E\). Let \((u, v)\) denote an unordered vertex pair of distinct vertices of \(G\). For a vertex \(u \in G,\) let \(N(u)\) be the set of all vertices of \(G\) which are adjacent to \(u\) in \(G.\) Then for \(0\leq i \leq n-1\), the \(i\)-equi neighbor set of \(G\) is defined as: \(N_{e}(G,i)=\{(u,v):u, v\in V, u\neq v\) and \(|N(u)|=|N(v)|=i\}.\) The equi-neighbor polynomial \(N_{e}[G;x]\) of \(G\) is defined as \(N_{e}[G;x]=\sum_{i=0}^{(n-1)} |N_{e}(G,i)| x^{i}.\) In this paper we discuss the equi-neighbor polynomial of graphs obtained by some binary graph operations.

Upper bounds for the multiplicative Y-index and S-index of some operations on graphs

Topological index is a numerical descriptor of a molecule; it is found that there is strong correlation between the proerties of chemical compounds and their molecular structure based on a specific topological feature of the corresponding molecular graph. In this paper, we introduce two new graph invariants known as the Multiplicative Y -index and Multiplicative S-index of a graph. We establish the upper bounds for the Multiplicative Y-index and Multiplicative S-index of the graph operations such as Join, Cartesian product, Composition, Tensor product, Strong product, Disjunction, Symmetric difference, Corona product, Corona join product and the indices are evaluated for some well-known graphs.

Embeddings of Graphs: Tessellate and Decussate Structures

We address the problem of finding a unique graph embedding that best describes a graph’s “topology” i.e., a canonical embedding (spatial graph). This question is of particular interest in the chemistry of materials. Graphs that admit a tiling in 3-dimensional Euclidean space are termed tessellate, those that do not decussate. We give examples of decussate and tessellate graphs that are finite and 3-periodic. We conjecture that a graph has at most one tessellate embedding. We give reasons for considering this the default “topology” of periodic graphs.

Heuristic Expanding Disconnected Graph: A Rapid Path Planning Method for Mobile Robots

Existing mobile robots mostly use graph search algorithms for path planning, which suffer from relatively low planning efficiency owing to high redundancy and large computational complexity. Due to the limitations of the neighborhood search strategy, the robots could hardly obtain the most optimal global path. A global path planning algorithm, denoted as EDG*, is proposed by expanding nodes using a well-designed expanding disconnected graph operator (EDG) in this paper. Firstly, all obstacles are marked and their corners are located through the map pre-processing. Then, the EDG operator is designed to find points in non-obstruction areas to complete the rapid expansion of disconnected nodes. Finally, the EDG* heuristic iterative algorithm is proposed. It selects the candidate node through a specific valuation function and realizes the node expansion while avoiding collision with a minimum offset. Path planning experiments were conducted in a typical indoor environment and on the public dataset CSM. The result shows that the proposed EDG* reduced the planning time by more than 90% and total length of paths reduced by more than 4.6%. Compared to A*, Dijkstra and JPS, EDG* does not show an exponential explosion effect in map size. The EDG* showed better performance in terms of path smoothness, and collision avoidance. This shows that the EDG* algorithm proposed in this paper can improve the efficiency of path planning and enhance path quality.

Extremal values for the spectral radius of the normalized distance Laplacian

The normalized distance Laplacian of a graph G is defined as DL(G)=T(G)−1/2(T(G)−D(G))T(G)−1/2 where D(G) is the matrix with pairwise distances between vertices and T(G) is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius.

Spectral extremal problem on the square of ℓ-cycle

Let Cℓ be the cycle of order ℓ. The square of Cℓ, denoted by Cℓ2, is obtained by joining all pairs of vertices with distance no more than two in Cℓ. Denote by ex(n,F) and spex(n,F) the maximum size and maximum spectral radius over all n-vertex F-free graphs, respectively. The well-known Turán problem asks for ex(n,F), and Nikiforov in 2010 proposed a spectral counterpart, known as the Brualdi-Solheid-Turán type problem, focusing on determining spex(n,F). In this paper, for any integer ℓ≥6 that is not divisible by 3, we characterize the unique extremal graph with respect to ex(n,Cℓ2) and spex(n,Cℓ2) for sufficiently large n, respectively.

Ordering of graphs with fixed size and diameter by A α-spectral radii

The A α -matrix of a graph G is defined as the convex linear combination of the adjacency matrix A ( G ) and the diagonal matrix of degrees D ( G ) , i.e. A α ( G ) = αD ( G ) + ( 1 − α ) A ( G ) with α ∈ [ 0 , 1 ] . The maximum modulus among all A α -eigenvalues is called the A α -spectral radius. In this paper, we order the connected graphs with size m and diameter (at least) d from the second to the ( ⌊ d 2 ⌋ + 1 ) th regarding to the A α -spectral radius for α ∈ [ 1 2 , 1 ) . As by-products, we identify the first ⌊ d 2 ⌋ largest trees of order n and diameter (at least) d in terms of their A α -spectral radii, and characterize the unique graph with at least one cycle having the largest A α -spectral radius among graphs of size m and diameter (at least) d. Consequently, the corresponding results for signless Laplacian matrix can be deduced as well.

Smaller kernels for two vertex deletion problems

In this paper we consider two vertex deletion problems. In the Block Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a block graph (a graph in which every biconnected component is a clique). In the Pathwidth One Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a graph with pathwidth at most one. We give a kernel for Block Vertex Deletion with O(k3) vertices and a kernel for Pathwidth One Vertex Deletion with O(k2) vertices. Our results improve the previous O(k4)-vertex kernel for Block Vertex Deletion (Agrawal et al., 2016 [1]) and the O(k3)-vertex kernel for Pathwidth One Vertex Deletion (Cygan et al., 2012 [3]).

Harary and hyper-Wiener indices of some graph operations

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring.

On the divisor graph of some graph operations

In this paper, we investigate when the generalized join product, generalized corona product, generalized edge corona product and the Kronecker product of graphs are divisor graphs. Also the situations under which the zero divisor graph and the co-maximal graph are divisor graphs are characterized.

The Laplacian spectrum of weighted composite networks and the applications

The topological properties of the networks can be described by the Laplacian spectra, but resolving the Laplacian spectra of networks poses difficulties. In this study, a novel approach for solving the Laplacian spectrum of weighted composite networks is presented. We first give the definitions of three weighted graph operations, namely, Cartesian product, corona, and join. Second, the Laplacian spectra of these composite networks are calculated. Finally, we use the obtained Laplacian spectrum to deduce some topological properties of the networks, such as network coherence, entire mean first-passage time, and Laplacian energy, which have several applications in physical chemistry.

General Position Polynomials

A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G. In this paper we introduce the general position polynomial as ∑i≥0aixi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{i \\ge 0} a_i x^i$$\\end{document}, where ai\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_i$$\\end{document} is the number of distinct general position sets of G with cardinality i. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs K(n, 2), with unimodal general position polynomials are presented.

Extremal spectral results of planar graphs without vertex‐disjoint cycles

AbstractGiven a planar graph family , let and be the maximum size and maximum spectral radius over all ‐vertex ‐free planar graphs, respectively. Let be the disjoint union of copies of ‐cycles, and be the family of vertex‐disjoint cycles without length restriction. Tait and Tobin determined that is the extremal spectral graph among all planar graphs with sufficiently large order , which implies the extremal graphs of both and for are . In this paper, we first determine and and characterize the unique extremal graph for , and sufficiently large . Second, we obtain the exact values of and , which solve a conjecture of Li for .

Leveraging Semantic Information for Enhanced Community Search in Heterogeneous Graphs

Community search (CS) is a vital research area in network science that focuses on discovering personalized communities for query vertices from graphs. However, existing CS methods mainly concentrate on homogeneous or simple attributed graphs, often disregarding complex semantic information and rich contents carried by entities in heterogeneous graphs (HGs). In this paper, we propose a novel problem, namely the “Semantic Network Oriented Community Search with Meta-Structures in Heterogeneous Graphs (SNCS),” which aims to find dense communities that contain the query vertex, with vertices of the same type sharing similar topics. In response to this new problem, we present a novel approach, also named SNCS, representing the first solution employing meta-structures and topic constraints to tackle community search, leveraging both topological and latent features. To overcome the high-time complexity challenge posed by searching through meta-structures, we introduce a unique graph reconstruction technique. Our proposed method’s superiority is validated through extensive evaluations on real-world datasets. The results demonstrate a significant improvement in the quality of the obtained communities, with increases of 3.5–4.4% in clustering coefficient and 5–11% in density while requiring only 4–46% of the running time when compared with the state-of-the-art methods.

Recognizing when a preference system is close to admitting a master list

A preference system I is an undirected graph where vertices have preferences over their neighbors, and I admits a master list if all preferences can be derived from a single ordering over all vertices. We study the problem of deciding whether a given preference system I is close to admitting a master list based on three different distance measures. We determine the computational complexity of the following questions: can I be modified by (i) k swaps in the preferences, (ii) k edge deletions, or (iii) k vertex deletions so that the resulting instance admits a master list? We investigate these problems in detail from the viewpoint of parameterized complexity and of approximation. We also present two applications related to stable and popular matchings.

Complexity of the (Connected) Cluster Vertex Deletion Problem on H-free Graphs

The well-known Cluster Vertex Deletion problem (cluster-vd) asks for a given graph G and an integer k whether it is possible to delete a set S of at most k vertices of G such that the resulting graph G-S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G-S$$\\end{document} is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs H for which cluster-vd on H-free graphs is polynomially solvable and for which it is NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-complete. Moreover, in the NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-completeness cases, cluster-vd cannot be solved in sub-exponential time in the vertex number of the H-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of cluster-vd, the Connected Cluster Vertex Deletion problem (connected cluster-vd), in which the set S has to induce a connected subgraph of G. It turns out that connected cluster-vd admits the same complexity dichotomy for H-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on H-free graphs.

On the Network Index of MAS with Layered Lattice-like Structures of Multiple Vertex-Related Parameters

In this article, a robust index named first-order network coherence (FONC) for the multi-agent systems (MASs) with layered lattice-like structure is studied via the angle of the graph spectra theory. The union operation of graphs is utilized to construct two pairs of non-isomorphic layered lattice-like structures, and the expression of the index is acquired by the approach of Laplacian spectra, then the corresponding asymptotic results are obtained. It is found that when the cardinality of the node sets of coronary substructures with better connectedness tends to infinity, the FONC of the whole network will have the same asymptotic behavior with the central lattice-like structure in the considered classic graph frameworks. The indices of the networks were simulated to illustrate the the asymptotic results, as described in the last section.

On the Entire Harmonic Index and Entire Harmonic Polynomial of Graphs

A topological descriptor is a numerical parameter that describes a chemical structure using the related molecular graph. Topological descriptors have significance in mathematical chemistry, particularly for studying QSPR and QSAR. In addition, if a topological descriptor has a reciprocal link with a molecular attribute, it is referred to as a topological index. The use of topological indices can help to examine the physicochemical features of chemical compounds because they encode certain attributes of a molecule. The Randić index is a molecular structure descriptor that has several applications in chemistry and medicine. In this paper, we introduce a new version of the Randić index to the inclusion of the intermolecular forces between bonds with atoms, referred to as an entire Harmonic index (EHI), and we present the entire Harmonic polynomial (EHP) of a graph. Specific formulas have been obtained for certain graph classes, and graph operations have been obtained. Bounds and some important results have been found. Furthermore, we demonstrate that the correlation coefficients for the new index lie between 0.909 and 1. In the context of enthalpy of formation and π-electronic energy, the acquired values are significantly higher than those observed for the Harmonic index and the Randić index.

Recursion Algorithm for Highlighting the Maximum Clique of the Graph

The work considers the polynomial algorithm for calculating the maximum clique of the non orientable graph. The algorithm is based on the properties of the subspace of sections and the subspace of cycles belonging to the the space sugrapha of a non — unoriented graph. It is shown that the algorithm for the allocation of the set of maximum clicks of the graph has an exponential computational complexity. Examples of highlighting the maximum clique of the graph are considered.