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Let D(G) and A(G), respectively, be the diagonal matrix of vertex de- grees and the adjacency matrix of a connected graph G. The generalized adjacency matrix of G is defined as Aα(G) = αD(G) + (1 − α)A(G), α ∈ [0, 1]. The spread of the generalized adjacency matrix, denoted by S(Aα(G)), is defined as the difference of the largest and smallest eigenvalues of Aα(G). The spread S(Q(G)) of Q(G), the signless Laplacian matrix of G and S(A(G)) of A(G) are similarly defined. In this paper, we establish the relationships between S(Aα(G)), S(Q(G)) and S(A(G)). Further, we find bounds for S(Aα(G)) involving different parameters of G. Also, we obtain lower bounds for S(Aα(G)) in terms of the clique number and independence number of G, and we characterize the extremal graphs in some cases. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce a new graph matrix, named the anti-adjacency matrix or eccentricity matrix, which is constructed from the distance matrix of a graph by keeping for each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping for each row and each column only the distances equal to 1. We show that the eccentricity matrix of trees is irreducible, and we investigate the relations between the eigenvalues of the adjacency and eccentricity matrices. Finally, we give some applications of this new matrix in terms of molecular descriptors, and we conclude by proposing some further research problems. [ABSTRACT FROM AUTHOR]

We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the g -Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved. [ABSTRACT FROM AUTHOR]

Abstract: Let be the smallest number such that the adjacency matrix of any undirected graph with vertices or more has at least nonpositive eigenvalues. We show that is well-defined and prove that the values of for are 1, 3, 6, 10, 16 respectively. In addition, we prove that for all , , in which is the Ramsey number for and , and is the triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the largest eigenvalue is bounded from below the largest degree, which generalizes some prior results. [Copyright &y& Elsevier]

The paper presents a novel spectral algorithm EVSA (eigenvector structure analysis), which uses eigenvalues and eigenvectors of the adjacency matrix in order to discover clusters. Based on matrix perturbation theory and properties of graph spectra we show that the adjacency matrix can be more suitable for partitioning than other Laplacian matrices. The main problem concerning the use of the adjacency matrix is the selection of the appropriate eigenvectors. We thus propose an approach based on analysis of the adjacency matrix spectrum and eigenvector pairwise correlations. Formulated rules and heuristics allow choosing the right eigenvectors representing clusters, i.e., automatically establishing the number of groups. The algorithm requires only one parameter-the number of nearest neighbors. Unlike many other spectral methods, our solution does not need an additional clustering algorithm for final partitioning. We evaluate the proposed approach using real-world datasets of different sizes. Its performance is competitive to other both standard and new solutions, which require the number of clusters to be given as an input parameter. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an "antisymmetrized adjacency matrix", specifically tailored for cycle digraphs (T 1 lattice) and simple directed paths (B 1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D -dimensional graph results from a Cartesian product of cycle digraphs (T 1 lattice) and simple directed paths (B 1 lattice). [ABSTRACT FROM AUTHOR]

This paper established the idea of minimum evdominating matrix of semigraph and calculated its energy. The minimum ev-dominating energy EmeD(G) of a semigraph G is the sum of the absolute values of the eigenvalues of the minimum ev-dominating matrix. Here some results are also derived in connection with the energy of minimum evdominating matrix. Some lower bounds are also established. [ABSTRACT FROM AUTHOR]

Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed. [ABSTRACT FROM AUTHOR]

This note is intended as a contribution to the study of quantitative measures of graph complexity that use entropy measures based on symmetry. Determining orbit sizes of graph automorphism groups is a key part of such studies. Here we focus on an extreme case where every orbit contains just a single vertex. A permutation of the vertices of a graph G is an automorphism if, and only if, the corresponding permutation matrix commutes with the adjacency matrix of G. This fact establishes a direct connection between the adjacency matrix and the automorphism group. In particular, it is known that if the eigenvalues of the adjacency matrix of G are all distinct, every non-trivial automorphism has order 2. In this note, we add a condition to the case of distinct eigenvalues that makes the graph asymmetric, i.e., reduces the automorphism group to the identity alone. In addition, we prove analogous results for the Google and Laplacian matrices. The condition is used to build an O(n3) algorithm for detecting identity graphs, and examples are given to demonstrate that it is sufficient, but not necessary. [ABSTRACT FROM AUTHOR]

Consider a simple, undirected graph G = (V;E), where A represents the adjacency matrix and Q represents the Laplacian matrix of G. The second smallest eigenvalue of Laplacian matrix of G is called the algebraic connectivity of G. In this article, we present a Python program for studying the Laplacian eigenvalues of a graph. Then, we determine the unique graph of minimum algebraic connectivity in the set of all tricyclic graphs. [ABSTRACT FROM AUTHOR]

Graph coarsening is a widely used dimensionality reduction technique for approaching large-scale graph machine-learning problems. Given a large graph, graph coarsening aims to learn a smaller-tractable graph while preserving the properties of the originally given graph. Graph data consist of node features and graph matrix (e.g., adjacency and Laplacian). The existing graph coarsening methods ignore the node features and rely solely on a graph matrix to simplify graphs. In this paper, we introduce a novel optimization-based framework for graph dimensionality reduction. The proposed framework lies in the unification of graph learning and dimensionality reduction. It takes both the graph matrix and the node features as the input and learns the coarsen graph matrix and the coarsen feature matrix jointly while ensuring desired properties. The proposed optimization formulation is a multi-block non-convex optimization problem, which is solved efficiently by leveraging block majorization-minimization, log determinant, Dirichlet energy, and regularization frameworks. The proposed algorithms are provably convergent and practically amenable to numerous tasks. It is also established that the learned coarsened graph is ϵ 2 (0, 1) similar to the original graph. Extensive experiments elucidate the efficacy of the proposed framework for real-world applications. The code for all the experimental results is available at CODE. [ABSTRACT FROM AUTHOR]

Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and distance matrix. We conclude by enumerating the prevelance of this construction in small graphs for the adjacency matrix, combinatorial Laplacian matrix, and distance Laplacian matrix. [ABSTRACT FROM AUTHOR]

Given a finite group G and the subgroups lattice L(G) of G, the \textit{non--permutability graph of subgroups} ΓL(G) is introduced as the graph with vertices in L(G)\CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and edges obtained by joining two vertices X,Y if XY≠YX. Here we study the behaviour of the non-permutability graph of subgroups using algebraic properties of associated matrices such as the adjacency and the Laplacian matrix. Further, we study the structure of some classes of groups whose non-permutability graph is strongly regular. [ABSTRACT FROM AUTHOR]

A Hamiltonian graph G is said to be hyper-Hamiltonian when G − v is Hamiltonian for any vertex v of G. In this paper, sufficient conditions for a graph to be hyper-Hamiltonian based on different parameters, such as degree and number of edges, are presented. Furthermore, other sufficient conditions for hyper-Hamiltonicity are obtained from spectral parameters, such as the spectral radii of the adjacency matrix, the signless Laplacian matrix and the distance matrix. Results on hyper-Hamiltonicity of a threshold graph through the eigenvalues of its Laplacian matrix are also presented. [ABSTRACT FROM AUTHOR]

The A γ matrix of a graph G is determined by A γ (G) = (1 − γ) A (G) + γ D (G) , where 0 ≤ γ ≤ 1 , A (G) and D (G) are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the A γ matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more γ adjacency-type matrices. In this paper, we obtain the A γ eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs. [ABSTRACT FROM AUTHOR]

Controllability is a fundamental issue in the field of fractional complex network control, yet it has not received adequate attention in the past. This paper is dedicated to exploring the controllability of complex networks involving the Caputo fractional derivative. By utilizing the Cayley–Hamilton theorem and Laplace transformation, a concise proof is given to determine the controllability of linear fractional complex networks. Subsequently, leveraging the Schauder Fixed-Point theorem, controllability Gramian matrix, and fractional calculus theory, we derive controllability conditions for nonlinear fractional complex networks with a weighted adjacency matrix and Laplacian matrix, respectively. Finally, a numerical method for the controllability of fractional complex networks is obtained using Matlab (2021a)/Simulink (2021a). Three examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Graph convolutional neural network architectures combine feature extraction and convolutional layers for hyperspectral image classification. An adaptive neighborhood aggregation method based on statistical variance integrating the spatial information along with the spectral signature of the pixels is proposed for improving graph convolutional network classification of hyperspectral images. The spatial-spectral information is integrated into the adjacency matrix and processed by a single-layer graph convolutional network. The algorithm employs an adaptive neighborhood selection criteria conditioned by the class it belongs to. Compared to fixed window-based feature extraction, this method proves effective in capturing the spectral and spatial features with variable pixel neighborhood sizes. The experimental results from the Indian Pines, Houston University, and Botswana Hyperion hyperspectral image datasets show that the proposed AN-GCN can significantly improve classification accuracy. For example, the overall accuracy for Houston University data increases from 81.71% (MiniGCN) to 97.88% (AN-GCN). Furthermore, the AN-GCN can classify hyperspectral images of rice seeds exposed to high day and night temperatures, proving its efficacy in discriminating the seeds under increased ambient temperature treatments. [ABSTRACT FROM AUTHOR]

A signed graph is a pair Γ = (G , σ) , where x = (V (G) , E (G)) is a graph and σ : E (G) → { + 1 , − 1 } is the sign function on the edges of G. For any α ∈ [ 0 , 1 ] we consider the matrix A α (Γ) = α D (G) + (1 − α) A (Γ) , where D (G) is the diagonal matrix of the vertex degrees of G , and A (Γ) is the adjacency matrix of Γ. Let m A α (Γ) (α) be the multiplicity of α as an A α (Γ) -eigenvalue, and let G have p (G) pendant vertices, q (G) quasi-pendant vertices, and no components isomorphic to K 2. It is proved that m A α (Γ) (α) = p (G) − q (G) whenever all internal vertices are quasi-pendant. If this is not the case, it turns out that m A α (Γ) (α) = p (G) − q (G) + m N α (Γ) (α) , where m N (Γ) (α) denotes the multiplicity of α as an eigenvalue of the matrix N α (Γ) obtained from A α (Γ) taking the entries corresponding to the internal vertices which are not quasi-pendant. Such results allow to state a formula for the multiplicity of 1 as an eigenvalue of the Laplacian matrix L (Γ) = D (G) − A (Γ). Furthermore, it is detected a class G of signed graphs whose nullity – i.e. the multiplicity of 0 as an A (Γ) -eigenvalue – does not depend on the chosen signature. The class G contains, among others, all signed trees and all signed lollipop graphs. It also turns out that for signed graphs belonging to a subclass G ′ ⊂ G the multiplicity of 1 as Laplacian eigenvalue does not depend on the chosen signatures. Such subclass contains trees and circular caterpillars. [ABSTRACT FROM AUTHOR]

Increasing evidences suggest that miRNAs play a key role in the occurrence and progression of many complex human diseases. Therefore, targeting dysregulated miRNAs with small molecule drugs in the clinical has become a new treatment. Nevertheless, it is high cost and time-consuming for identifying miRNAs-targeted with drugs by biological experiments. Thus, more reliable computational method for identification associations of drugs with miRNAs urgently need to be developed. In this study, we proposed an efficient method, called GNMFDMA, to predict potential associations of drug with miRNA by combining graph Laplacian regularization with non-negative matrix factorization. We first calculated the overall similarity matrices of drugs and miRNAs according to the collected different biological information. Subsequently, the new drug-miRNA association adjacency matrix was reformulated based on the K nearest neighbor profiles so as to put right the false negative associations. Finally, graph Laplacian regularization collaborative non-negative matrix factorization was used to calculate the association scores of drugs with miRNAs. In the cross validation, GNMFDMA obtains AUC of 0.9193, which outperformed the existing methods. In addition, case studies on three common drugs (i.e., 5-Aza-CdR, 5-FU and Gemcitabine), 30, 31 and 34 of the top-50 associations inferred by GNMFDMA were verified. These results reveal that GNMFDMA is a reliable and efficient computational approach for identifying the potential drug-miRNA associations. [ABSTRACT FROM AUTHOR]

Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L (D) = D e g + (D) − A (D) and Q (D) = D e g + (D) + A (D) , where A (D) represents the adjacency matrix and D e g + (D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations. [ABSTRACT FROM AUTHOR]

Defining a sound shift operator for graph signals, similar to the shift operator in classical signal processing, is a crucial problem in graph signal processing (GSP), since almost all operations, such as filtering, transformation, prediction, are directly related to the graph shift operator. We define a set of energy-preserving shift operators that satisfy many properties similar to their counterparts in classical signal processing, but are different from the shift operators defined in the literature, such as the graph adjacency matrix and Laplacian matrix based shift operators, which modify the energy of a graph signal. We decouple the graph structure represented by eigengraphs and the eigenvalues of the adjacency matrix or the Laplacian matrix. We show that the adjacency matrix of a graph is indeed a linear shift invariant (LSI) graph filter with respect to the defined shift operator. We further define autocorrelation and cross-correlation functions of signals on the graph, enabling us to obtain the solution to the optimal filtering on graphs, i.e., the corresponding Wiener filtering on graphs and the efficient spectra analysis and frequency domain filtering in parallel with those in classical signal processing. This new shift operator based GSP framework enables the signal analysis along a correlation structure defined by a graph shift manifold as opposed to classical signal processing operating on the assumption of the correlation structure with a linear time shift manifold. Several illustrative simulations are presented to validate the performance of the designed optimal LSI filters. [ABSTRACT FROM PUBLISHER]

Graph entropy is an important measure of the evolution and complexity of networks. Bipartite graph is a special network and an important mathematical model for system resource allocation and management. In reality, a network system usually has obvious directionality. The direction of the network, or the movement trend of the network, can be described with spectrum index. However, little research has been done on the eigenvalue-based entropy of directed bipartite network. In this study, based on the adjacency matrix, the in-degree Laplacian matrix and the in-degree signless Laplacian matrix of directed bipartite graph, we defined the eigenvalue-based entropy for the directed bipartite network. Using the eigenvalue-based entropy, we described the evolution law of the directed bipartite network structure. Aiming at the direction and bipartite feature of the directed bipartite network, we improved the generation algorithm of the undirected network. We then constructed the directed bipartite nearest-neighbor coupling network, directed bipartite small-world network, directed bipartite scale-free network, and directed bipartite random network. In the proposed model, spectrum of those directed bipartite network is used to describe the directionality and bipartite property. Moreover, eigenvalue-based entropy is empirically studied on a real-world directed movie recommendation network, in which the law of eigenvalue-base entropy is observed. That is, if eigenvalue-based entropy value of the recommendation system is large, the evolution of movie recommendation network becomes orderless. While if eigenvalue-based entropy value is small, the structural evolution of the movie recommendation network tends to be regular. The simulation experiment shows that eigenvalue-based entropy value in the real directed bipartite network is between the values of a directed bipartite small world and a scale-free network. It shows that the real directed bipartite network has the structural property of the two typical directed bipartite networks. The coexistence of the small-world phenomena and the scale-free phenomena in the real network is consistent with the evolution law of typical network models. The experimental results show that the validity and rationality of the definition of eigenvalue-based entropy, which serves as a tool in the analysis of directed bipartite networks. [ABSTRACT FROM AUTHOR]

Signal processing on graphs finds applications in many areas. In recent years, renewed interest on this topic was kindled by two groups of researchers. Narang and Ortega constructed two-channel filter banks on bipartitie graphs described by Laplacians. Sandryhaila and Moura developed the theory of linear systems, filtering, and frequency responses for the case of graphs with arbitrary adjacency matrices, and showed applications in signal compression, prediction, etc. Inspired by these contributions, this paper extends classical multirate signal processing ideas to graphs. The graphs are assumed to be general with a possibly nonsymmetric and complex adjacency matrix. The paper revisits ideas, such as noble identities, aliasing, and polyphase decompositions in graph multirate systems. Drawing such a parallel to classical systems allows one to design filter banks with polynomial filters, with lower complexity than arbitrary graph filters. It is shown that the extension of classical multirate theory to graphs is nontrivial, and requires certain mathematical restrictions on the graph. Thus, classical noble identities cannot be taken for granted. Similarly, one cannot claim that the so-called delay chain system is a perfect reconstruction system (as in classical filter banks). It will also be shown that $M$ -partite extensions of the bipartite filter bank results will not work for $M$-channel filter banks, but a more restrictive condition called $M$-block cyclic property should be imposed. Such graphs are studied in detail. A detailed theory for $M$ -channel filter banks is developed in a companion paper. [ABSTRACT FROM PUBLISHER]

In this article, we propose a new type of square matrix associated with an undirected graph by trading off the natural embedded symmetry in them. The proposed matrix is defined using the neighbourhood sets of the vertices, called as neighbourhood matrixNM(G). The proposed matrix also exhibits a bijection between the product of the two graph matrices, namely the adjacency matrix and the graph Laplacian. Alternatively, we define this matrix by using the breadth-first search traversals from every vertex, and the subgraph induced by the first two levels in the level decomposition from that vertex. The two levels in the level decomposition of the graph give us more information about the neighbours along with the neighbours-of-neighbour of a vertex. This insight is required and is found useful in studying the impact of broadcasting on social networks, in particular, and complex networks, in general. We establish several properties of NM(G). Additionally, we also show how to reconstruct a graph G, given an NM(G). The proposed matrix also solvesmany graph-theoretic problems using less time complexity in comparison to the existing algorithms. [ABSTRACT FROM AUTHOR]

Using the notions of clique partitions and edge clique covers of graphs, we consider the corresponding incidence structures. This connection furnishes lower bounds on the negative eigenvalues and their multiplicities associated with the adjacency matrix, bounds on the incidence energy, and on the signless Laplacian energy for graphs. For the more general and well-studied set S (G) of all real symmetric matrices associated with a graph G, we apply an extended version of an incidence matrix tied to an edge clique cover to establish several classes of graphs that allow two distinct eigenvalues. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices and let λ1 ⩾ λ2 ⩾ ‣ ⩾ λn be the eigenvalues of its adjacency matrix. For random graphs we investigate the sum of eigenvalues s k = ∑ i = 1 k λ i for 1 ⩾ k ⩾ n, and show that a typical graph has Sk ⩾ (e(G) + k2)/(0.99n)1/2, where e(G) is the number of edges of G. We also show bounds for the sum of eigenvalues within a given range in terms of the number of edges. The approach for the proofs was first used in Rocha (2020) to bound the partial sum of eigenvalues of the Laplacian matrix. [ABSTRACT FROM AUTHOR]

Abstract: The Randić index R of a graph G is defined as the sum of over all edges of G, where denotes the degree of a vertex in G. is the largest eigenvalue of the signless Laplacian matrix of G, where D is the diagonal matrix with degrees of the vertices on the main diagonal and A is the adjacency matrix of G. Hansen and Lucas [18] conjectured (1) and equality holds for and (2) with equality if and only if for and for , respectively. In this paper, we prove the conjecture (1) and obtain a result very close to the conjecture (2). Moreover, we give some results relating harmonic index and the largest eigenvalue of the adjacency matrix. [Copyright &y& Elsevier]

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A (G) and D (G) and defined the A α -matrix for every real α ∈ [ 0 , 1 ] as: A α (G) = α D (G) + (1 − α) A (G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the A α -matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established. [ABSTRACT FROM AUTHOR]

Graph learning from data is a canonical problem that has received substantial attention in the literature. Learning a structured graph is essential for interpretability and identification of the relationships among data. In general, learning a graph with a specific structure is an NP-hard combinatorial problem and thus designing a general tractable algorithm is challenging. Some useful structured graphs include connected, sparse, multi-component, bipartite, and regular graphs. In this paper, we introduce a unified framework for structured graph learning that combines Gaussian graphical model and spectral graph theory. We propose to convert combinatorial structural constraints into spectral constraints on graph matrices and develop an optimization framework based on block majorization-minimization to solve structured graph learning problem. The proposed algorithms are provably convergent and practically amenable for a number of graph based applications such as data clustering. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. An open source R package containing the code for all the experiments is available at https://CRAN.R-project.org/package=spectralGraphTopology. [ABSTRACT FROM AUTHOR]

The energy of a simple graph G , E ( G ) , is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E ( L G ) ( L G is the line graph of G ) and LE + ( G ) , respectively. In this paper we obtain a relation between E ( L G ) and LE + ( G ) of graph G . From this relation we characterize all the graphs satisfying E ( L G ) = LE + ( G ) + 4 m n − 4 . We also present a relation between E ( G ) and E ( L G ) . Moreover, we give an upper bound on E ( L G ) of graph G and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

We compute the elementary divisors of the adjacency and Laplacian matrices of families of polar graphs. These graphs have as vertices the isotropic one-dimensional subspaces of finite vector spaces with respect to non-degenerate forms, with adjacency given by orthogonality. [ABSTRACT FROM AUTHOR]

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of either +1 or −1. This generalizes signed graphs to a hypergraph setting and simultaneously provides a natural definition for a signed adjacency which is used to define the adjacency and Laplacian matrices. Many properties of these matrices are known, but there are no nontrivial families of oriented hypergraphs with their spectrum determined. In this paper we define and study hypergraph families that are analogous to cycles and paths in graphs and signed graphs. The adjacency and Laplacian eigenvalues for some of these families is determined. [ABSTRACT FROM AUTHOR]

The second largest eigenvalue of a graph is an important algebraic parameter which is related with the expansion, connectivity and randomness properties of a graph. Expanders are highly connected sparse graphs. In coding theory, Expander codes are Error Correcting codes made up of bipartite expander graphs. In this paper, first we prove the interlacing of the eigenvalues of the adjacency matrix of the bipartite graph with the eigenvalues of the bipartite quotient matrices of the corresponding graph matrices. Then we obtain bounds for the second largest and second smallest eigenvalues. Since the graph is bipartite, the results for Laplacian will also hold for Signless Laplacian matrix. We then introduce a new method called vertex-split of a bipartite graph to construct asymptotically good expander codes with expansion factor D/2 < α < D and ϵ < 1/2 and prove a condition for the vertex-split of a bipartite graph to be k-connected with respect to λ2. Further, we prove that the vertex-split of G is a bipartite expander. Finally, we construct an asymptotically good expander code whose factor graph is a graph obtained by the vertex-split of a bipartite graph. [ABSTRACT FROM AUTHOR]

Protein–protein interaction (PPI) networks consist of the physical and/or functional interactions between the proteins of an organism, and they form the basis for the field of network medicine. Since the biophysical and high-throughput methods used to form PPI networks are expensive, time-consuming, and often contain inaccuracies, the resulting networks are usually incomplete. In order to infer missing interactions in these networks, we propose a novel class of link prediction methods based on continuous-time classical and quantum walks. In the case of quantum walks, we examine the usage of both the network adjacency and Laplacian matrices for specifying the walk dynamics. We define a score function based on the corresponding transition probabilities and perform tests on six real-world PPI datasets. Our results show that continuous-time classical random walks and quantum walks using the network adjacency matrix can successfully predict missing protein–protein interactions, with performance rivalling the state-of-the-art. [ABSTRACT FROM AUTHOR]

Voice activity detection remains a significant challenge in the presence of transients since transients are more dominant than speech, though it has achieved satisfactory performance in quasi-stationary noisy environments. This paper studies the differences between speech and transients in nonlinear dynamic characteristics and proposes a new method for accurately detecting speech and transients. Limited by algorithm complexity, previous research has proposed few detectors to model speech and transients based on contextual information and thus failing to detect transient frames accurately. To address this challenge, our study proposes to map features of audio signals to a time series complex network, a kind of graph data, analyzed by the Laplacian and adjacency matrix of graphs, then classified by the support vector machine (SVM) classifier. The proposed algorithm can analyze a more extended speech period, allowing the full utilization of contextual information of preceding and following frames. The experimental results show that the performance of this method has obvious superiority over other existing algorithms. [ABSTRACT FROM AUTHOR]

Entropy is an important index for describing the structure, function, and evolution of network. The existing research on entropy is primarily applied to undirected networks. Compared with an undirected network, a directed network involves a special asymmetric transfer. The research on the entropy of directed networks is very significant to effectively quantify the structural information of the whole network. Typical complex network models include nearest-neighbour coupling network, small-world network, scale-free network, and random network. These network models are abstracted as undirected graphs without considering the direction of node connection. For complex networks, modeling through the direction of network nodes is extremely challenging. In this paper, based on these typical models of complex network, a directed network model considering node connection in-direction is proposed, and the eigenvalue entropies of three matrices in the directed network is defined and studied, where the three matrices are adjacency matrix, in-degree Laplacian matrix and in-degree signless Laplacian matrix. The eigenvalue-based entropies of three matrices are calculated in directed nearest-neighbor coupling, directed small world, directed scale-free and directed random networks. Through the simulation experiment on the real directed network, the result shows that the eigenvalue entropy of the real directed network is between the eigenvalue entropy of directed scale-free network and directed small-world network. [ABSTRACT FROM AUTHOR]

Abstract: We give sharp upper and lower bounds for the spectral radius of a nonnegative matrix with all row sums positive using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We compare these bounds with the known bounds using the row sums by examples. We also apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix and some distance-based matrices. Some known results are generalized and improved. [Copyright &y& Elsevier]

Abstract: We give upper and lower bounds for the spectral radius of a nonnegative matrix using its row sums and characterize the equality cases if the matrix is irreducible. Then we apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix. Some known results in the literature are generalized and improved. [Copyright &y& Elsevier]

The harmonic index of a conected graph G is defined as H(G) = uv2E(G) 2 d(u)+d(v), where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G)+A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have...and the bounds are best possible. [ABSTRACT FROM AUTHOR]

In this study we consider connected graphs of fixed order n and size n that minimize the largest eigenvalue of the adjacency matrix, also known as the spectral radius. Such graphs are called minimizers. The motivation for this research lies in the fact that the spectral radius plays a significant role in modelling virus propagation in complex networks, in the sense that a smaller spectral radius ensures a better virus protection in the network modelled by the corresponding graph. We conjecture that vertex degrees of a minimizer are as equal as possible, i.e. belong to {⌊2m∕m⌋, ⌈2m∕m⌉}. The conjecture is confirmed for graphs with at most 10 vertices by the total enumeration, and some classes of graphs are resolved theoretically. In general, exact determining of minimizers is quite difficult and computationally exhaustive, and thus we propose a long-scale variable neighbourhood search as an alternative approach. We employ this heuristic search for selected instances concerning graphs with at most 100 vertices, and as a result we always obtain a graph with required vertex degrees. The search efficiently results in solutions that are (in a precise sense) close to the optimal ones, i.e. to minimizers. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph of order n and size m. The matrix L(G) = D(G)-A(G) is the Laplacian matrix of G, where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. For the graph G, let d1 ≥ d2 ≥ ··· dn be the vertex degree sequence and μ1 ≥ ·2 ≥ ··· ≥ μn-1 > μn = 0 be the Laplacian eigenvalues. The Laplacian resolvent energy RL(G) of a graph G is defined as RL(G) = Pn i=1 1 n+1-μi. In this paper, we obtain an upper bound for the Laplacian resolvent energy RL(G) in terms of the order, size and the algebraic connectivity of the graph. Further, we establish relations between the Laplacian resolvent energy RL(G) with each of the Laplacian-energy-Like invariant LEL, the Kirchhoff index Kf and the Laplacian energy LE of the graph. [ABSTRACT FROM AUTHOR]

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies approximating the eigenvalues of massively large versions of these two aforementioned matrices. The proposed solutions outlined in this paper essentially are a reformulated multiple linear regression analysis for the first problem and a matrix inertia refinement adapted to existing work for the second problem. [ABSTRACT FROM AUTHOR]

Studying the association of the brain's structure and function with neurocognitive outcomes requires a comprehensive analysis that combines different sources of information from a number of brain-imaging modalities. Recently developed regularization methods provide a novel approach using information about brain structure to improve the estimation of coefficients in the linear regression models. Our proposed method, which is a special case of the Tikhonov regularization, incorporates structural connectivity derived with Diffusion Weighted Imaging and cortical distance information in the penalty term. Corresponding to previously developed methods that inform the estimation of the regression coefficients, we incorporate additional information via a Laplacian matrix based on the proximity measure on the cortical surface. Our contribution consists of constructing a principled formulation of the penalty term and testing the performance of the proposed approach via extensive simulation studies and a brain-imaging application. The penalty term is constructed as a weighted combination of structural connectivity and proximity between cortical areas. Simulation studies mimic the real brain-imaging settings. We apply our approach to the study of data collected in the Human Connectome Project, where the cortical properties of the left hemisphere are found to be associated with vocabulary comprehension. 1. Introduction [ABSTRACT FROM AUTHOR]

Let G be a complex unit gain graph which is obtained from an undirected graph Γ by assigning a complex unit φ ( v i v j ) to each oriented edge v i v j such that φ ( v i v j ) φ ( v j v i ) = 1 for all edges. The Laplacian matrix of G is defined as L ( G ) = D ( G ) − A ( G ) , where D ( G ) is the degree diagonal matrix of Γ and A ( G ) = ( a i j ) has a i j = φ ( v i v j ) if v i is adjacent to v j and a i j = 0 otherwise. In this paper, we provide a combinatorial description of det ( L ( G ) ) that generalizes that for the determinant of the Laplacian matrix of a signed graph. [ABSTRACT FROM AUTHOR]

For a finite group G and a subset X≠ 0 of G, the commuting graph, indicated by G = C(G, X), is the simple connected graph with vertex set X and two distinct vertices x and y are edge connected in G if and only if they commute in X. The Aα matrix of G is specified as Aα(G) = αD(G) + (1-α)A(G), α ∈ [0, 1], where D(G) is the diagonal matrix of vertex degrees while A(G) is the adjacency matrix of G. In this article, we investigate the Aα matrix for commuting graphs of finite groups and we also find the Aα eigenvalues of the dihedral, the semidihedral and the dicyclic groups. We determine the upper bounds for the largest Aα eigenvalue for these graphs. Consequently, we get the adjacency eigenvalues, the Laplacian eigenvalues, and the signless Laplacian eigenvalues of these graphs for particular values of α. Further, we show that these graphs are Laplacian integral. [ABSTRACT FROM AUTHOR]

Given a regular (connected) graph Γ = (X, E) with adjacency matrix A, d+1 distinct eigenvalues, and diameter D, we give a characterization of when its distance matrix AD is a polynomial in A, in terms of the adjacency spectrum of Γ and the arithmetic (or harmonic) mean of the numbers of vertices at distance at most D − 1 from every vertex. The same result is proved for any graph by using its Laplacian matrix L and corresponding spectrum. When D = d we reobtain the spectral excess theorem characterizing distance-regular graphs. [ABSTRACT FROM AUTHOR]

We consider the critical group of a hypothetical Moore graph of diameter 2 and valency 57 . Determining this group is equivalent to finding the Smith normal form of the Laplacian matrix of such a graph. We show that all of the Sylow p -subgroups of the critical group must be elementary abelian with the exception of p = 5 . We prove that the 5 -rank of the Laplacian matrix determines the critical group up to two possibilities. [ABSTRACT FROM AUTHOR]

Clustering large hyperspectral images (HSIs) is a very challenging problem because large HSIs have high dimensionality, large spectral variability, and large computational and memory consumption. Recently, sparse subspace clustering (SSC) has achieved remarkable success in HSI clustering. However, most SSC-based methods suffer from the following bottlenecks for large HSIs: 1) high computational consumption and memory space during the construction of the similarity matrix and decomposition of the graph Laplacian matrix and 2) failure to capture the relationships among dictionary atoms, sparse coefficients, and hyperspectral pixels. To address these challenges, we propose a novel algorithm that extends SSC to cocluster large HSIs, called bipartite graph partition with graph nonnegative matrix factorization (BGP-GNMF). Specifically, to fully explore the characteristics of the spectral and spatial contexts in HSIs, we propose a novel superpixel and pixel coclustering framework with bipartite graph partitioning in the joint sparse representation domain, where superpixel-based dictionary atoms are defined as disjoint vertex sets of the bipartite graph and the joint sparsity representation is mapped into the adjacency matrix of the undirected bipartite graph. To overcome the challenges of high computational consumption and large memory space for large HSIs, the bipartite graph partition with orthonormal constrained nonnegative matrix factorization is proposed to simultaneously cluster the structured dictionary atoms and hyperspectral pixels with an indicator matrix. Finally, to exploit the intrinsic geometry of HSIs, we incorporate manifold regularization into the bipartite graph partition to improve final clustering accuracy. The effectiveness and efficiency of the proposed method are verified on three classical HSIs, and the experimental results illustrate the superiority of the proposed method compared with other state-of-the-art HSI clustering methods. [ABSTRACT FROM AUTHOR]