Spectral Graph Theory Research Articles

On the eccentricity spectra of chain graphs

On the eccentricity spectra of chain graphs

Cokernel statistics for walk matrices of directed and weighted random graphs

Abstract The walk matrix associated to an $n\times n$ integer matrix $\mathbf{X}$ and an integer vector $b$ is defined by ${\mathbf{W}} \,:\!=\, (b,{\mathbf{X}} b,\ldots, {\mathbf{X}}^{n-1}b)$ . We study limiting laws for the cokernel of $\mathbf{W}$ in the scenario where $\mathbf{X}$ is a random matrix with independent entries and $b$ is deterministic. Our first main result provides a formula for the distribution of the $p^m$ -torsion part of the cokernel, as a group, when $\mathbf{X}$ has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the ${\mathbb{Z}}[x]$ -module structure. The motivation for this work arises from an open problem in spectral graph theory, which asks to show that random graphs are often determined up to isomorphism by their (generalised) spectrum. Sufficient conditions for generalised spectral determinacy can, namely, be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper.

Sequence of Bounds for Spectral Radius and Energy of Digraph

The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph D having at least one doubly adjacent vertex in terms of indegree is proposed. Particularly, it is exhibited that ρ(D)≥αj=∑p=1m(χj+1(p))2∑p=1m(χj(p))2, such that equality is attained iff D=G↔+ {DE∉ Cycle}, where each component of associated graph is a k-regular or (k1,k2) semiregular bipartite. By utilizing the sequence of lower bounds of the spectral radius of D, the sequence of upper bounds of energy of D, where the sequence decreases when eU≤αj and increases when eU>αj, are also proposed. All of the obtained inequalities are elaborated using examples. We also discuss the monotonicity of these sequences.

THE GENERAL SYMMETRIC DEGREE AND GENERAL SYMMETRIC DEGREE ADJACENCY POLYNOMIAL OF A GRAPH AND ITS ENERGY

The spectral graph theory is concerned with the relationship between the spectra of specific matrices associated with a graph and the structural properties of that graph. This paper introduces a general symmetric degree adjacency of a graph G, general symmetric degree of a graph G, and complete weighted graph of a graph. Here we explore its characteristic polynomial using Sachs theorem. We also investigate the bounds for index and energy of general symmetric degree adjacency of a graph. We give two algorithms, first one to find general symmetric degree adjacency of a graph G and second one to find general symmetric degree of a graph G.

The number of spanning trees in K n -chain and ring graphs

The problem of counting spanning trees of graphs or networks is a fundamental and crucial area of research in combinatorics, while has numerous important applications in statistical physics, network theory and theoretical computer science. Very recently, Kosar, Zaman, Ali and Ullah obtained a nice formula on the number of spanning trees of a K 5-chain network K5ℓ constructed by connecting ℓcopys of complete graphs K 5. They made extensive use of matrix theory and spectral graph theory, especially the normalized Laplacian of graphs. In this paper, by using a rather simple and more physical treatment (the mesh-star transformation in electrical network) without any linear algebra, we generalize their result to K n -chain graphs and K n -ring graphs. The results show that there is a simple relation between the number of spanning trees of the K n -chain graph Lnℓ and the K n -ring graph Cnℓ . We also calculate the corresponding tree entropy (or so called ‘the asymptotic growth constant’) and find that the tree entropy of the corresponding K n -chain graphs and K n -ring graphs are totally the same.

Bayesian inference of frequency-specific functional connectivity in MEG imaging using a spectral graph model

Abstract Understanding the relationship between structural connectivity (SC) and functional connectivity (FC) of the human brain is an important goal of neuroscience. Highly detailed mathematical models of neural masses exist that can simulate the interactions between functional activity and structural wiring. These models are often complex and require intensive computation. Most importantly, they do not provide a direct or intuitive interpretation of this structure–function relationship. In this study, we employ the emerging concepts of spectral graph theory to obtain this mapping in terms of graph harmonics, which are eigenvectors of the structural graph’s Laplacian matrix. In order to imbue these harmonics with biophysical underpinnings, we leverage recent advances in parsimonious spectral graph modeling (SGM) of brain activity. Here, we show that such a model can indeed be cast in terms of graph harmonics, and can provide a closed-form prediction of FC in an arbitrary frequency band. The model requires only three global, spatially invariant parameters, yet is capable of generating rich FC patterns in different frequency bands. Only a few harmonics are sufficient to reproduce realistic FC patterns. We applied the method to predict FC obtained from pairwise magnitude coherence of source-reconstructed resting-state magnetoencephalography (MEG) recordings of 36 healthy subjects. To enable efficient model inference, we adopted a deep neural network-based Bayesian procedure called simulation-based inference. Using this tool, we were able to speedily infer not only the single most likely model parameters, but also their full posterior distributions. We also implemented several other benchmark methods relating SC to FC, including graph diffusion and coupled neural mass models. The present method was shown to give the best performance overall. Notably, we discovered that a single biophysical parameterization is capable of fitting FCs from all relevant frequency bands simultaneously, an aspect that did not receive adequate attention in prior computational studies.

Rényi–Sobolev Inequalities and Connections to Spectral Graph Theory

Rényi–Sobolev Inequalities and Connections to Spectral Graph Theory

Vertex coupling interpolation in quantum chain graphs

We analyze the band spectrum of the periodic quantum graph in the form of a chain of rings connected by line segments with the vertex coupling which violates the time reversal invariance, interpolating between the δ coupling and the one determined by a simple circulant matrix. We find that flat bands are generically absent and that the negative spectrum is nonempty even for interpolation with a non-attractive δ coupling; we also determine the high-energy asymptotic behavior of the bands.

Optimizing network insights: AI-Driven approaches to circulant graph based on Laplacian spectra

The study of Laplacian and signless Laplacian spectra extends across various fields, including theoretical chemistry, computer science, electrical networks, and complex networks, providing critical insights into the structures of real-world networks and enabling the prediction of their structural properties. A key aspect of this study is the spectrum-based analysis of circulant graphs. Through these analyses, important network measures such as mean-first passage time, average path length, spanning trees, and spectral radius are derived. This research enhances our understanding of the relationship between graph spectra and network characteristics, offering a comprehensive perspective on complex networks. Consequently, it supports the ability to make predictions and conduct analyses across a wide range of scientific disciplines.

On spectral extrema of graphs with given order and generalized 4-independence number

Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph G, a vertex subset S is called a maximum generalized 4-independent set of G if the induced subgraph G[S] dose not contain a 4-tree as its subgraph, and the subset S has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of G. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the n-vertex graphs having the minimum spectral radius with generalized 4-independence number ψ, where ψ⩾⌈3n/4⌉. Finally, we identify all the connected n-vertex graphs with generalized 4-independence number ψ∈{3,⌈3n/4⌉,n−1,n−2} having the minimum spectral radius.

Complete equitable decompositions

A classical result in spectral graph theory states that if a graph G has an equitable partition π then the eigenvalues of the divisor graph Gπ are a subset of its eigenvalues, i.e. σ(Gπ)⊆σ(G). A natural question is whether it is possible to recover the remaining eigenvalues σ(G)−σ(Gπ) in a similar manner. Here we show that any weighted undirected graph with nontrivial equitable partition can be decomposed into a number of subgraphs whose collective spectra contain these remaining eigenvalues. Using this decomposition, which we refer to as a complete equitable decomposition, we introduce an algorithm for finding the eigenvalues of an undirected graph (symmetric matrix) with a nontrivial equitable partition. Under mild assumptions on this equitable partition we show that we can find eigenvalues of such a graph faster using this method when compared to standard methods. This is potentially useful as many real-world data sets are quite large and have a nontrivial equitable partition.

Hyper-Zagreb index in fuzzy environment and its application

The Zagreb indices (ZIs) are important graph invariants that are used extensively in many different fields in mathematics and chemistry, such as network theory, spectral graph theory, fuzzy graph theory (FGT) and molecular chemistry, etc. The hyper-ZI is introduced especially for fuzzy graphs (FGs) in this study. The study computes this index's bounds for a variety of FG types, including paths, cycles, stars, complete FGs and partial fuzzy subgraphs. It is shown that isomorphic FGs produce the same values for this index. Moreover, interesting connections are established between the hyper-ZI and the second ZI for FGs. Moreover, bounds on this index are found for the following operations: direct product, Cartesian product, composition, join, union, strong product and semi-strong product of two FGs. In the end, the effectiveness of this index is compared with three other topological indices: hyper-ZI for crisp graphs, first ZI for FGs and F-index for FGs, in an analysis of the crime “Murder” in India. While the hyper-ZI for FGs, first ZI for FGs and F-index for FGs yield similar outcomes, the hyper-ZI for FGs demonstrates superior realism in detecting crimes in India compared to its crisp graph counterpart.

The inertia bound is far from tight

AbstractThe inertia bound and ratio bound (also known as the Cvetković bound and Hoffman bound) are two fundamental inequalities in spectral graph theory, giving upper bounds on the independence number of a graph in terms of spectral information about a weighted adjacency matrix of . For both inequalities, given a graph , one needs to make a judicious choice of weighted adjacency matrix to obtain as strong a bound as possible. While there is a well‐established theory surrounding the ratio bound, the inertia bound is much more mysterious, and its limits are rather unclear. In fact, only recently did Sinkovic find the first example of a graph for which the inertia bound is not tight (for any weighted adjacency matrix), answering a longstanding question of Godsil. We show that the inertia bound can be extremely far from tight, and in fact can significantly underperform the ratio bound: for example, one of our results is that for infinitely many , there is an ‐vertex graph for which even the unweighted ratio bound can prove , but the inertia bound is always at least . In particular, these results address questions of Rooney, Sinkovic, and Wocjan–Elphick–Abiad.

On the Asymptotic Network Indices of Weighted Three-Layered Structures with Multi-Fan Composed Subgraphs

In this paper, three sorts of network indices for the weighted three-layered graph are studied through the methods of graph spectra theory combined with analysis methods. The concept of union of graphs are applied to design two sorts of weighted layered multi-fan composed graphs, and the accurate mathematical expressions of the network indices are obtained through the derivations of Laplacian spectra; furthermore, the asymptotic properties are also derived. We find that when the cardinalities of the vertices on a sector-edge-link tend to infinity, the indices of FONC and EMFPT are irrelevant with the number of copies of the fan-substructure based on the considered graph framework.

On Some Distance Spectral Characteristics of Trees

Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study of graphs with “few eigenvalues” is a contemporary problem in spectral graph theory. This paper studies graphs with few distinct distance eigenvalues. After mentioning the classification of graphs with one and two distinct distance eigenvalues, we mainly focus on graphs with three distinct distance eigenvalues. Characterizing graphs with three distinct distance eigenvalues is “highly” non-trivial. In this paper, we classify all trees whose distance matrix has precisely three distinct eigenvalues. Our proof is different from earlier existing proof of the result as our proof is extendable to other similar families such as unicyclic and bicyclic graphs. The main tools which we employ include interlacing and equitable partitions. We also list all the connected graphs on ν ≤ 6 vertices and compute their distance spectra. Importantly, all these graphs on ν ≤ 6 vertices are determined from their distance spectra. We deliver a distance cospectral pair of order 7, thus making it a distance cospectral pair of the smallest order. This paper is concluded with some future directions.

Solving Mazes: A New Approach Based on Spectral Graph Theory

The use of graph theory for solving labyrinths and mazes is well known, understanding the possible paths as the connections between the nodes that represent the corners or bifurcations. This work presents a new idea: minimizing the length of the optimal path formulated as a topology optimization problem. The maze is mapped with finite elements, and then, through the eigenvalues of the Laplacian matrix of the graph, a constraint is imposed over the connectivity between the input and the output. Several 2D examples are provided to support this approach, allowing for unequivocally finding the shortest path in mazes with multiple connections between entrance and exit, resulting in an nonheuristic algorithm.

Geometric spectral theory of quantum graphs

Geometric spectral theory of quantum graphs

Accelerating structure search using atomistic graph-based classifiers.

We introduce an atomistic classifier based on a combination of spectral graph theory and a Voronoi tessellation method. This classifier allows for the discrimination between structures from different minima of a potential energy surface, making it a useful tool for sorting through large datasets of atomic systems. We incorporate the classifier as a filtering method in the Global Optimization with First-principles Energy Expressions (GOFEE) algorithm. Here, it is used to filter out structures from exploited regions of the potential energy landscape, whereby the risk of stagnation during the searches is lowered. We demonstrate the usefulness of the classifier by solving the global optimization problem of two-dimensional pyroxene, three-dimensional olivine, Au12, and Lennard-Jones LJ55 and LJ75 nanoparticles.

On spectral radius of generalized arithmetic–geometric matrix of connected graphs and its applications

The generalized arithmetic–geometric matrix of a simple connected graph G=(V,E) is defined to be the |V|×|V| matrix whose ij-entry is diα+djα2diαdjα if vivj∈E, and 0 otherwise, where α is an arbitrary real number and di is the degree of vi∈V. This matrix is a general form of the arithmetic–geometric matrix (α=1) and the extended adjacency matrix (α=2), both of which have been well studied in spectral graph theory and chemical graph theory. In this paper, we focus on the spectral radius ρagα of the generalized arithmetic–geometric matrix of connected graphs. Some chemical applications of ρagα are explored, and some extremal results on ρagα are obtained; in particular, the connected graphs (including trees) having the maximum and minimum ρagα are determined.

Automatic construction and optimization method of enterprise data asset knowledge graph based on graph attention network

Traditional knowledge graph construction methods often rely on a large amount of human intervention and specialized knowledge, which seems inefficient and error-prone when dealing with massive, multi-source, and heterogeneous data assets of enterprises. To address this issue, a Multi Neighborhood Awareness Network (MNAN) model was proposed, which combines a multi language pre training model based on Bidirectional Encoder Representations from Transformers (BERT) and a Graph Convolutional Neural Network (GCN). By introducing an improved attention mechanism to deeply explore the neighborhood characteristics of entities, the accuracy and efficiency of entity matching are enhanced. In addition, to improve the integrity of the knowledge graph, a dynamic graph attention network (RDGAT) model based on graph attention network fusion relationship features is proposed. Through dynamic attention mechanism and relationship fusion strategy, deep learning of relationship features is carried out to optimize the completion performance of the knowledge graph spectrum. The performance test results show that the Hits@1 metric of the multi-neighborhood perceptual network model is close to 90% as the proportion of aligned entity seeds increases to 50%; the average inverse rank of the dynamic graph attention network on the test dataset improves by 6.3%. The experimental results show that the research-proposed knowledge graph fusion and complementation model exhibits significant effectiveness in automatically identifying and integrating large-scale, multi-source heterogeneous data at the enterprise level, and can significantly improve the automation level and accuracy of knowledge graph construction. The research method has achieved significant results in the data asset knowledge graph of petroleum enterprises and other fields, providing strong support for applications such as intelligent search, recommendation systems, and semantic analysis.