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Language phrases are an effective way to express uncertain pieces of information, and easily conforms to the language habits of decision makers to describe the evaluation of things. The consistency judgment of a linguistic judgment matrices is the key to analytic hierarchy process (AHP). If a linguistic judgment matrix has a satisfactory consistency, then the rank of the decision schemes can be determined. In this study, the comparison relation between the decision schemes is first represented by a directed graph. The preference relation matrix of the linguistic judgment matrix is the adjacency matrix of the directed graph. We can use the n −1 st power of the preference relation to judge the linguistic judgment matrix whether has a satisfactory consistency. The method is utilized if there is one and only one element in the n −1 st power of the preference relation, and the element 1 is not on the main diagonal. Then the linguistic judgment matrix has a satisfactory consistency. If there are illogical judgments, the decision schemes that form a 3-loop can be identified and expressed through the second-order sub-matrix of the preference relation matrix. The feasibility of this theory can be verified through examples. The corresponding schemes for illogical judgments are represented in spatial coordinate system. [ABSTRACT FROM AUTHOR]

The research on node importance identification for temporal networks has attracted much attention. In this work, combined with the multi-layer coupled network analysis method, an optimized supra-adjacency matrix (OSAM) modeling method was proposed. In the process of constructing an optimized super adjacency matrix, the intra-layer relationship matrixes were improved by introducing the edge weight. The inter-layer relationship matrixes were formed by improved similarly and the inter-layer relationship is directional by using the characteristics of directed graphs. The model established by the OSAM method accurately expresses the structure of the temporal network and considers the influence of intra- and inter-layer relationships on the importance of nodes. In addition, an index was calculated by the average of the sum of the eigenvector centrality indices for a node in each layer and the node importance sorted list was obtained from this index to express the global importance of nodes in temporal networks. The experimental results on three real temporal network datasets Enron, Emaildept3, and Workspace showed that compared with the SAM and the SSAM methods, the OSAM method has a faster message propagation rate and larger message coverage and better SIR and NDCG@10 indicators. [ABSTRACT FROM AUTHOR]

Let Gσ be an oriented graph with underlying simple graph G. The skew-adjacency matrix of Gσ is the {0, 1,-1}-matrix S = S(Gσ) = [sij ], such that sij = 1 if (vi, vj) is an arc in Gσ, sij = -1 if (vj, vi) is an arc in Gσ and sij = 0, otherwise. In this paper, all connected oriented graphs with three distinct skew-eigenvalues 0 and ±2i are characterized. [ABSTRACT FROM AUTHOR]

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ ( G ) ≥ 3 , then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D 1 is a primitive regular orientation and D 2 is a non-regular orientation of a given graph, then for sufficiently large t , the number of closed walks of length t in D 1 is more than the number of closed walks of length t in D 2 . [ABSTRACT FROM AUTHOR]

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the (i, j) entry of Am (A is adjacency matrix) is equal to the number of walks of length m from vertex i to vertex j, we show that elements of mth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers. [ABSTRACT FROM AUTHOR]

Let J(λ;n1,…,nk) be the set of matrices A such that λ is an eigenvalue of A and n1⩽⋯⩽nk are the sizes of the Jordan blocks associated with λ. For a given index v of A, denote by A−v the principal submatrix of co-order one obtained from A by deleting the vth row and column. In the present paper, all possible changes of the part of the Jordan form corresponding to λ under the transition from A to A−v are determined for matrices A∈J(λ;n1,…,nk) such that for the eigenvalue λ of both A and A⊤, there exists a Jordan chain of the largest length nk whose eigenvector has nonzero vth entry. In particular, it is shown that for almost every matrix A∈J(λ;n1,…,nk), n1,…,nk−1 are the sizes of Jordan blocks for λ considered as an eigenvalue of A−v. Moreover, it is also proved that if A is the adjacency matrix of a vertex-transitive digraph and k⩾2, then the change n1,…,nk→n1,…,nk−2,2nk−1−1 holds for the eigenvalue λ under the transition from A to A−v. In the case of k=1, λ is a simple eigenvalue of A and does not belong to the spectrum of A−v. [Copyright &y& Elsevier]

Under elimination of two arcs, spectrum of digraph having ring structure and two Hamiltonian cycles is shown real, if and only if the number of vertices is even and “interval between arcs,” eliminated from the same cycle, is maximal. Possibility to apply the obtained results to evaluate failure-resistance of networks with ring topology is considered. [ABSTRACT FROM AUTHOR]

Deza digraphs were introduced in 2003 by Zhang and Wang as directed graph version of Deza graphs, that also generalize the notion of directed strongly regular graphs. In this paper, we give several new constructions of Deza digraphs. Further, we introduce twin and Siamese twin (directed) Deza graphs and construct several examples. Finally, we study a variation of directed Deza graphs and provide a construction from finite fields. [ABSTRACT FROM AUTHOR]

This work surveys the use of directed weighted graphs in conducting comparative static analyses. The paper discusses the implementation of a computer-aided process for building spreadsheet-based graph models for inter- and intra-regional financial flows. The graph-theoretic techniques are programmed to enable the interactive visualization and analysis of financial data using Wolfram technologies (i.e., Mathematica software v. 11.3 or later, Wolfram player v. 12 or later). The paper describes the workflow for several interactive visualizations applicable to financial networks. The author provides four programs, written in the Wolfram language, that customize input–output financial models by combining the Manipulate command with built-in Mathematica functions and functions of the IGraph package (IGraph/M). The study is a tutorial article for the generation of a suite of visual schemes that provide patterns, practices, and roadmaps of the financial markets across the globe. [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]

Let G be a simple graph, and ⃗G be an oriented graph of G with an orientation and skew-adjacency matrix S(⃗ G). Let λ1(⃗ G), λ2(⃗ G), . . ., λn(⃗ G) be the eigenvalues of S(⃗ G). The number Pn i=1 λki (⃗ G) (k = 0, 1, . . ., n - 1), denoted by Tk(⃗ G), is called the k-th skew spectral moment of ⃗ G, and T(⃗ G) = (T0(⃗ G), T1(⃗ G), . . ., Tn-1(⃗ G)) is the sequence of skew spectral moments of ⃗ G. Suppose ⃗G 1 and ⃗G 2 are two digraphs. We shall write ⃗G 1 ≺T ⃗G 2 (⃗G 1 comes before ⃗G 2 in a T-order) if for some k (1 ≤ k ≤ n-1), Ti(⃗G 1) = Ti(⃗G 2) (i = 0, 1, . . ., k-1) and Tk(⃗ G1) < Tk(⃗G 2) hold. For two given positive integers p and q with p ≤ q, we denote T p,q n = {T: T is a tree of order n with a (p, q)-bipartition }. In this paper, we discuss T-order among all trees in T p,q n . Furthermore, the last three trees, in the T-order, underlying graphs among T p,q n (4 ≤ p ≤ q) are characterized. [ABSTRACT FROM AUTHOR]

In an earlier work, the author together with Guo [7] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory. [ABSTRACT FROM AUTHOR]

Abstract It is proved that for any finite connected graph G , there exists an orientation of G such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of G (with equality if and only if G is a tree). This provides a suitable answer to a problem proposed by Mohar. The proof uses the method of interlacing families of polynomials that was developed by Marcus, Spielman, and Srivastava in their seminal work on the existence of infinite families of Ramanujan graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study 24 possibilities of directed strongly regular graphs with adjacency matrix of rank 6, our proof is based on two effective algorithms. At last, we get 4 families of directed strongly regular graphs with realizable parameter sets, exclude 20 families of directed strongly regular graphs with feasible parameter sets; partially solve the classification problem of directed strongly regular graphs with adjacency matrix of rank 6. [ABSTRACT FROM AUTHOR]

In order to realize "from individual data research to data system research" and "from passive data verification to active discovery," this study proposes a hypergraph-based association rule redundancy processing algorithm in data mining. This study introduces the concepts of hypergraph and system, explores the establishment of hypergraph on a three-dimensional matrix model, and adopts a new hyperedge definition method according to the characteristics of big data and the concept of the system, which improves the ability to deal with problems; the association rules are transformed into a directed hypergraph, and the adjacency matrix is redefined. The detection of redundancy and loops is transformed into the processing of connected blocks and cycles in the hypergraph. The experimental results show that two UCI datasets were selected, namely, the balloons dataset and the shuttle landing control dataset, in which the minimum support and minimum confidence of the balloons dataset are both 5%. The dataset has 4 attributes, and 18 association rules are obtained through the Aprior algorithm. Although the running time of the coevolution algorithm is slightly longer than that of the other two global optimization algorithms, the running time is completely within the acceptable range. Moreover, due to the effective introduction of the idea of coevolution, compared with the use of the other two algorithms for association rule mining, it not only has a better mining quality but also has a significant advantage in the ability to jump out of the local optimal solution, realizing the search of high-quality association rules in high-dimensional datasets. Conclusion. This model provides a new idea and method for the redundant processing of association rules. [ABSTRACT FROM AUTHOR]

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]

Abstract: A weighted directed graph is a directed graph G whose underlying undirected graph is simple and whose edges have nonzero (directional) complex weights, that is, the presence of an edge of weight is as good as the presence of the edge with weight , the complex conjugate of . Let G be a weighted directed graph on vertices . Denote by the weight of an edge . The adjacency matrix of G is an matrix with entries or or 0, depending on whether or or otherwise, respectively. We supply a characterization of those unicyclic weighted directed graphs G whose edges have weights from the set and whose adjacency matrix satisfies the following property: ‘λ is an eigenvalue of with multiplicity k if and only if is an eigenvalue of with the same multiplicity’. [Copyright &y& Elsevier]

Chimeric antigen receptor T-cell (CAR-T) immunotherapy, a novel approach for treating blood cancer, is associated with the production of cytokine release syndrome (CRS), which poses significant safety concerns for patients. Currently, there is limited knowledge regarding CRS-related cytokines and the intricate relationship between cytokines and cells. Therefore, it is imperative to explore a reliable and efficient computational method to identify cytokines associated with CRS. In this study, we propose Meta-DHGNN, a directed and heterogeneous graph neural network analysis method based on meta-learning. The proposed method integrates both directed and heterogeneous algorithms, while the meta-learning module effectively addresses the issue of limited data availability. This approach enables comprehensive analysis of the cytokine network and accurate prediction of CRS-related cytokines. Firstly, to tackle the challenge posed by small datasets, a pre-training phase is conducted using the meta-learning module. Consequently, the directed algorithm constructs an adjacency matrix that accurately captures potential relationships in a more realistic manner. Ultimately, the heterogeneous algorithm employs meta-photographs and multi-head attention mechanisms to enhance the realism and accuracy of predicting cytokine information associated with positive labels. Our experimental verification on the dataset demonstrates that Meta-DHGNN achieves favorable outcomes. Furthermore, based on the predicted results, we have explored the multifaceted formation mechanism of CRS in CAR-T therapy from various perspectives and identified several cytokines, such as IFNG (IFN-γ), IFNA1, IFNB1, IFNA13, IFNA2, IFNAR1, IFNAR2, IFNGR1 and IFNGR2 that have been relatively overlooked in previous studies but potentially play pivotal roles. The significance of Meta-DHGNN lies in its ability to analyze directed and heterogeneous networks in biology effectively while also facilitating CRS risk prediction in CAR-T therapy. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset . It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group is normal iff is either abelian or has the form for some non-negative integer , where is the quaternion group and is the abelian group of order and exponent 2. [Copyright &y& Elsevier]

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 A (1 , ≤ ℓ) -identifying code in digraph D is a dominating subset C of vertices of D , such that all distinct subsets of vertices of D with cardinality at most ℓ have distinct closed in-neighborhoods within C. As far as we know, it is the very first time that the spectral graph theory has been applied to the identifying codes. We give a new method to obtain an upper bound on ℓ for digraphs. The results obtained here can also be applied to graphs. [ABSTRACT FROM AUTHOR]

Abstract: Let be a simple graph of order with triangle(s). Also let be the eigenvalues of the adjacency matrix of graph . X. Yong [X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73–80] conjectured that (i) is complete if and only if and also (ii) is complete if and only if . Here we disprove this conjecture by a counter example. Wang et al. [J.F. Wang, F. Belardo, Q.X. Huang, B. Borovićanin, On the two largest Q-eigenvalues of graphs, Discrete Math. 310 (2010) 2858–2866] conjectured that friendship graph is determined by its adjacency spectrum. Here we prove this conjecture. The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean value of eccentricities of all vertices of . Moreover, we mention three conjectures, obtained by the system AutoGraphiX, about the average eccentricity , girth and the spectral radius of graphs (see Aouchiche (2006) [1], available online at http://www.gerad.ca/~agx/). We give a proof of one conjecture and disprove two conjectures by counter examples. [Copyright &y& Elsevier]

Abstract: The problem of identifying those simple, undirected graphs with vertices and edges that have the smallest minimum eigenvalue of the adjacency matrix is considered. Several general properties of the minimizing graphs are described. These strongly suggest bipartition, to the extent possible for the number of edges. In the bipartite case, the precise structure of the minimizing graphs is given for a number of infinite classes. [Copyright &y& Elsevier]

Abstract: Let be a function. Let be the matrix defined by if for some i and j and otherwise. We describe the Jordan canonical form of the matrix in terms of the directed graph for which is the adjacency matrix. We discuss several examples including a connection with the Collatz conjecture. [Copyright &y& Elsevier]

We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented. [Copyright &y& Elsevier]

In this article, a general approach for directed graph clustering and two new density-based clustering objectives are presented. First, using an equivalence between the clustering objective functions and a trace maximization expression, the directed graph clustering objectives are converted into the corresponding weighted kernel $k$ -means problems. Then, a nonspectral algorithm, which covers both the direction and weight information of the directed graphs, is thus proposed. Next, with Rayleigh’s quotient, the upper and lower bounds of clustering objectives are obtained. After that, we introduce a new definition of weak links to characterize the effectiveness of clustering. Finally, illustrative examples are given to demonstrate effectiveness of the results. This article provides a glance at the potential connection between density-based and pattern-based clustering. Compared with other approaches for directed graph clustering, the method proposed in this article naturally avoids the loss of the nonsymmetric edge data because there is no need for any additional symmetrization. [ABSTRACT FROM AUTHOR]

The flywheel energy storage (FES) array system plays an important role in smoothing the power output of wind farms. Therefore, how to allocate the total charging and discharging power of wind farms to individual FES system (FESS) units has long been a research hotspot. We propose a hierarchical coordinated control strategy applying the concept of the state of energy (SOE) of FESS units. Energy allocation is achieved by investigating the consensus algorithm of the SOE for both an undirected graph and unbalanced directed graph. Then, we propose another approach to convert the adjacency matrix into a column stochastic matrix using the idea of the PageRank algorithm for an unbalanced directed graph and achieve an SOE consensus of FESS units using the power iteration method. Simulation results demonstrate that the proposed algorithm is effective. [ABSTRACT FROM AUTHOR]

Abstract: The skew energy of a digraph D is defined as the sum of the singular values of its skew adjacency matrix . In this paper, we first interpret the entries of the power of the skew adjacency matrix of a digraph in terms of the number of its walks and then focus on the question posed by Adiga et al. [C. Adiga, R. Balakrishnan, Wasin So, The skew energy of a graph, Linear Algebra Appl. 432 (2010) 1825–1835] of determining all -regular connected digraphs with optimum skew energy. [Copyright &y& Elsevier]

A mathematical model is proposed in order to obtain an automatized tool to remove any unnecessary data, to compute the level of the redundancy, and to recover the original and filtered database, at any time of the process, in a vector database. This type of database can be modeled as an oriented directed graph. Thus, the database is characterized by an adjacency matrix. Therefore, a record is no longer a row but a matrix. Then, the problem of cleaning redundancies is addressed from a theoretical point of view. Superficial redundancy is measured and filtered by using the 1-norm of a matrix. Algorithms are presented by Python and MapReduce, and a case study of a real cybersecurity database is performed. [ABSTRACT FROM AUTHOR]

Let G be a digraph of order n. The energy of G is the sum of absolute values of the real parts of the eigenvalues of its adjacency matrix. The monad graph is a digraph pertaining to the finite group G, where every vertex of the elements g of G is adjacent to its image by a directed edge under the action of the map f(g) = g². In this paper, we compute the energy of monad graphs. [ABSTRACT FROM AUTHOR]

Fix a constant C ≥ 1 and let d = d(n) satisfy d ≤ lnC n for every large integer n. Denote by An the adjacency matrix of a uniform random directed d-regular graph on n vertices. We show that if d → ∞ as n → ∞, the empirical spectral distribution of the appropriately rescaled matrix An converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in a directed d -regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of An based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between matrix entries. [ABSTRACT FROM AUTHOR]

How spontaneous brain neural activities emerge from the underlying anatomical architecture, characterized by structural connectivity (SC), has puzzled researchers for a long time. Over the past decades, much effort has been directed toward the graph modeling of SC, in which the brain SC is generally considered as relatively invariant. However, the graph representation of SC is unable to directly describe the connections between anatomically unconnected brain regions and fail to model the negative functional correlations. Here, we extend the static graph model to a spatiotemporal varying hypergraph Laplacian diffusion (STV-HGLD) model to describe the propagation of the spontaneous neural activity in human brain by incorporating the Laplacian of the hypergraph representation of the structural connectome (${h}$ SC) into the regular wave equation. Theoretical solution shows that the dynamic functional couplings between brain regions fluctuate in the form of an exponential wave regulated by the spatiotemporal varying Laplacian of ${h}$ SC. Empirical study suggests that the cortical wave might give rise to resonance with SC during the self-organizing interplay between excitation and inhibition among brain regions, which orchestrates the cortical waves propagating with harmonics emanating from the ${h}$ SC while being bound by the natural frequencies of SC. Besides, the average statistical dependencies between brain regions, normally defined as the functional connectivity (FC), arises just at the moment before the cortical wave reaches the steady state after the wave spreads across all the brain regions. Comprehensive tests on four extensively studied empirical brain connectome datasets with different resolutions confirm our theory and findings. [ABSTRACT FROM AUTHOR]

We study spectral properties of the Manhattan products of the path graphs and show the concentration of zero-eigenvelues. [ABSTRACT FROM AUTHOR]

A graph, consisting of undirected edges, can be represented as a sum of two digraphs, consisting of oppositely oriented directed edges. Gutman and Plath in [J. Serb. Chem. Soc. 66 (2001), 237-241] showed that for annulenes, the eigenvalue spectrum of the graph is equal to the sum of the eigenvalue spectra of respective two digraphs. Here we exhibit a number of other graphs with this property. [ABSTRACT FROM AUTHOR]

Let π = { V 1 , ... , V r } be an equitable partition of the vertex set of a directed graph (digraph) X. It is well known that the characteristic polynomial ϕ (X / π , x) of a quotient graph X / π divides that of X , but the remainder part is not well investigated. In this paper, we define a deletion graph X \ π over an equitable partition π , which is a signed directed graph defined for a fixed set of deleting vertices { v ‾ i ∈ V i , i = 1 , ⋯ , r } , and give a similarity transformation exchanging the adjacency matrix A (X) which is compatible with the equitable partition for a block triangular matrix whose diagonal blocks are the adjacency matrix of the quotient graph and the deletion graph. In fact, we show the result for more general matrices including adjacency matrix of graphs, and as corollaries, we show the followings: (i) a decomposition formula of the reciprocal of the Ihara-Bartholdi zeta function over an equitably partitioned undirected graph into the quotient graph part and the deletion graph part, and (ii) Chen and Chen's result ([4, Theorem 3.1]) on the Ihara-Bartholdi zeta functions on generalized join graphs, and (iii) Teranishi's result [32, Theorem 3.3]. [ABSTRACT FROM AUTHOR]

Given the adjacency matrix A of a digraph, the eigenvalues of the matrix AAT constitute the so-called non-negative spectrum of this digraph. We investigate the relation between the structure of digraphs and their non-negative spectra and associated eigenvectors. In particular, it turns out that the non-negative spectrum of a digraph can be derived from the traditional (adjacency) spectrum of certain undirected bipartite graphs. [ABSTRACT FROM AUTHOR]

Let D be an oriented graph, i.e., a digraph in which all arcs are not symmetric, with vertex set { 1 , ... , n }. The adjacency matrix A = (a i j) of D is the n × n matrix defined as a i j = 1 if ij is an arc of D and a i j = 0 if ij is not an arc of D. The rank of D is defined to be the rank of its adjacency matrix. In this paper, oriented graphs of rank 2 are characterized. [ABSTRACT FROM AUTHOR]

Let D be a digraph, and let Ln(D) and k(D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k(Ln(D))=k(D)=0 if every connected component of D is a directed cycle. (2) k(Ln(D))=k(D)+n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that if D has no sources or no sinks then k(D)&les;k(L(D))&les;k(D)+1. [Copyright &y& Elsevier]

Motivated by the theory of Cuntz-Krieger algebras we define and study C^\ast-algebras associated to directed quantum graphs. For classical graphs the C^\ast-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to KK-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C^\ast-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of KK-theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general. [ABSTRACT FROM AUTHOR]

In this study, dynamic formation tracking and interception are performed by controlling multi-agent using a Euler-like Lagrangian model. The purpose is to use the distance-based rigid graph method to control multi-agent, and ultimately achieve dynamic formation tracking and target interception of multi-agent. Initially, distance-based graph stiffness and back-stepping techniques were considered to address the formation control challenge. This method helps achieve the initial expected formation and effectively complete the formation mission. Leaders continue to chase the moving target, while followers stick to the expected arrangement. The leader then precisely tracks the moving target and surrounds it within the formation. By using Lyapunov stability theory with adaptive control, it is ensured that the total distance meets finite and consistent error limits. Finally, the numerical simulation of the interception plan was carried out by 6 multi-agents and 1 target at different times to verify the effectiveness of the control method. [ABSTRACT FROM AUTHOR]

Network representation learning aims to map the relationship between network nodes and context nodes to a low-dimensional representation vector space. Directed network representation learning considers mapping directional of node vector. Currently, only sporadic work on direct network representation has been reported. In this work, we propose a novel algorithm that takes into account the direction of the edge with text's attribute of the node in directed network representation learning. We then define the matrix based on in-degree of Laplacian and signless Laplacian for digraph, and it utilizes web page datasets from universities in the USA to evaluate the performance of vertex classification. We compare our algorithm with other directed representation learning algorithms. The experimental results show that our algorithm outperforms the baseline by over 20% when the training ratio ranges from 10 to 90%. We apply the in-degree-Laplacian and In-degree-signless-Laplacian to directed representation learning, which is one of the main contributions of this algorithm. Additionally, we incorporate text information through matrix completion in directed network representation learning and the experimental results show an increase in performance of up to 20% compared to the baseline, especially when the training ratio is 10%. [ABSTRACT FROM AUTHOR]

Currently, autonomous underwater vehicles (AUVs) are facing various challenges, rendering multiple-AUV (multi-AUV) formation control a pivotal research direction. The issues surrounding formation control for a multi-AUV system to establish time-varying formations must be investigated. This paper discusses the formation protocol of multi-AUV systems in order to establish the defined time-varying formations. First, when these systems establish formations, the speed of each AUV can be equivalent. After that, consensus-based methods are used to solve the time-varying formation-control problem. The necessary and sufficient process of multi-AUV in achieving time-varying formations is proved. Furthermore, the formula for the time-varying formation center function is provided. Further, we present a protocol law for multi-AUVs to establish time-varying formations. Finally, the theoretical results of a simulation are presented, which validate the formation protocol. [ABSTRACT FROM AUTHOR]

Operation Sequencing (OS) is one of the most critical tasks in a CAPP system. This process could be modelled as a combinatorial problem where finding a suitable solution within a reasonable time interval is difficult. This work implements a novel Discrete Differential Evolution Algorithm (DDEA) to solve the OS problem, focusing on parts of up to 76 machining operations; the relationships among operations are represented as a directed graph; the contributions of the DDEA are as follows: (1) operates with a discrete representation in the space of feasible solutions; (2) employs mutation and crossover operators to update solutions and to reduce machining and setup costs, (3) possess a local search strategy to achieve better solutions, and (4) integrates a statistical method based on quantiles to measure the quality and likelihood for an achieving a solution. To demonstrate the efficiency and robustness of the DDEA, five prismatic parts with different numbers of machining operations as benchmarks to address the OS problem were selected. The results generated the same OS for parts with a few machining operations (up to 23 machining operations). Conversely, for parts with more machining operations, the DDEA needs more runs to achieve the best solution. [ABSTRACT FROM AUTHOR]

An oriented graph G σ is a digraph without loops and multiple arcs, where G is the underlying graph of G σ . Let S ( G σ ) denote the skew-adjacency matrix of G σ , and A ( G ) be the adjacency matrix of G . The rank (resp. skew-rank) of G (resp. G σ ), written as r ( G ) (resp. s r ( G σ ) ), refers to the rank of A ( G ) (resp. S ( G σ ) ). It is natural and interesting to study the relationship between s r ( G σ ) and r ( G ) . Wong, Ma and Tian (European J. Combin. 54 (2016) 76–86) [30] determined the sharp upper bound on s r ( G σ ) − r ( G ) . As a continuance of it, in this paper, a sharp lower bound on s r ( G σ ) − r ( G ) is determined; as well a sharp upper bound on s r ( G σ ) / r ( G ) is determined. All the corresponding extremal oriented graphs G σ are characterized, respectively. [ABSTRACT FROM AUTHOR]

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. The iota energy of a digraph is recently defined as the sum of absolute values of imaginary part of its eigenvalues. In this paper, we extend the concept of iota energy of digraphs to weighted digraphs. We compute the iota energy formulae for the positive and negative weight directed cycles. We also characterize the unicyclic weighted digraphs with cycle weight r ∈ [-1, 1]\{0} having minimum and maximum iota energy. We obtain well known McClelland upper bound for the iota energy of weighted digraphs. Finally, we find the class of noncospectral equienergetic weighted digraphs. [ABSTRACT FROM AUTHOR]

It is well-known that the paths are determined by the spectrum of the adjacency matrix. For digraphs, every digraph whose underlying graph is a tree is cospectral to its underlying graph with respect to the Hermitian adjacency matrix ( H -cospectral). Thus every (simple) digraph whose underlying graph is isomorphic to P n is H -cospectral to P n . Interestingly, there are others. This paper finds digraphs that are H -cospectral with the path graph P n and whose underlying graphs are nonisomorphic, when n is odd, and finds that such graphs do not exist when n is even. In order to prove this result, all digraphs whose Hermitian spectral radius is smaller than 2 are determined. [ABSTRACT FROM AUTHOR]