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The collection of SuperDARN ionospheric echo data to make ionospheric convection maps is of great significance for Space Weather research. The ionospheric echoes for SuperDARN are generally mixed with scatter from the ground or sea surface, thus the clustering analysis of SuperDARN backscatter echoes is important. In this paper, the first implementation of deep clustering based on the graph automatic encoder network is efficiently realized for classifying the SuperDARN data. In addition, the model is compared with the traditional algorithm and machine learning clustering algorithms. Application of the model to sample data reveals that the deep clustering algorithm can capture the structural characteristics of the echoes and improve the accuracy of echo clustering and classifying. [ABSTRACT FROM AUTHOR]

Introduced by Gutman & Trinajstić in 1972, the two well-known Zagreb indices efficiently determine the total π -electron energy of alternant hydrocarbons. The hyper-Zagreb index is a variant of the classical first Zagreb index. A tree is an acyclic graphical structure. A segment in a tree is equivalent to a vertex of degree two. This paper derives sharp upper and lower bounds on the hyper-Zagreb index of trees with given number of segments. We also characterize the corresponding trees achieving those bounds. The second part of the paper studies the predictive potential of the hyper-Zagreb index for determining physicochemical properties such as the enthalpies of combustion, formation, sublimation, and vaporization of monocarboxylic acids. Our statistical analysis asserts that the hyper-Zagreb index correlates these physicochemical properties of monocarboxylic acids with correlation coefficient ρ > 0.99. The strongest correlation by the hyper-Zagreb index was obtained with the enthalpy of combustion having correlation coefficient ρ = 0.999991. This warrants the further employability of the hyper-Zagreb index in QSPR modeling. This potential applicability justifies the construction of the hyper-Zagreb graphical index whose motivation of defining was purely mathematical. [ABSTRACT FROM AUTHOR]

A neural network is a computer system modeled on the nerve tissue and nervous system. In this sense, neural networks refer to systems of neurons, either organic or artificial in nature. Neural networks have diverse applications in computer graphics, artificial intelligence (AI), machine & deep learning, chemistry, material science, among others. Research on structural properties and their classification of favorable or unfavorable for neural networks has been started recently. Employing tools from mathematics and specifically graph theory has been a key research direction in this area. Different graph-theoretic parameters have potential applications in studying topological properties of neural networks. In this paper for the first time, we study some nondeterministic polynomial-time (NP)-complete and NP-hard problems on various neural networks by considering their graphs. We consider classes of 3- and 4-layered probabilistic neural networks (PNNs), cellular neural networks (CNNs), and Tickysym spiking neural networks (TSNNs). In order to study their topological properties, we study their structures of maximal cliques, minimal colorings, maximal independent sets, maximal and perfect matchings, and minimum dominating sets. Computational results on the clique number, chromatic number, independence number, matching ratio and the domination number are reported. For instance, the clique number of the probabilistic neural networks is 2, the Tickysym spiking neural network is 3, whereas, for the cellular neural network, it is 4. These numerical results quantify the incoming/outgoing traffic in these architectures. Similarly, the independence number of 25-nodal 3-layered probabilistic neural network is 15, whereas, the independence number of 25-nodal cellular neural network is just 9. Moreover, the asymptotic matching ratio for PNNs (resp. CNNs) is 1 2 (resp. 1 8 ), whereas, it is 1 6 for the case of TSNNs. This gives probabilistic neural networks a significant priority in controllability over cellular neural networks. Some open problems are proposed in the end to further this research area. [ABSTRACT FROM AUTHOR]

The extraction of the communities is reflected by the allocation of the nodes to the different communities and therefore a distribution, to which we can associate a score indicating its quality. Several methods have been proposed to extract existing communities in a network. The classification of these methods may be the subject of a separate study. The common point noted between most of these methods is the use of a function that measures the quality of a distribution. A metric called "Modularity" is considered to be the most efficient; it is used in most of the community extraction algorithms. Other metrics have been proposed such as "Z-Modularity", "Modularity density", "Exponential modularity". However, Modularity is still a competing metric. Our job is to establish an in-depth study of these quality measurement functions. For this purpose, considering the case of non-overlapping communities, we thought of a structural comparison between the results obtained, executing the "Leiden" algorithm, each time we change the metric among the four adopted. We chose this algorithm because it is an improved version of one of the most popular algorithms in this area called "Louvain". [ABSTRACT FROM AUTHOR]

Data-driven approaches have been recognized as a new paradigm for establishing and exploring process-morphology-property relationships. However, typical exploration methods deliver high-dimensional morphologies that pose the challenge of extracting the key features and patterns that could guide the processing and materials design. The high dimensionality also hampers the organization of the data and the associated data analytics. As a solution, the currently available approaches either take a simplified view of the morphology, e.g., focusing on pixels in the morphology images, or apply transformations that average out structural descriptors of morphologies. To address these shortcomings, we propose a new computationally efficient and configurable distance operator that takes an intermediate approach. Our main idea is to represent the morphology as a graph where graph connectivity reflects the relative arrangement of components (e.g., grains, droplets) in the morphology, and the label of the graph vertices captures the domain-specific information of each characteristic domain. Next, given the graph abstraction, the distance between morphologies is computed using vectorized graph-based representation. Because both morphology graph structure and associated signature functions have clear interpretations, our distance measure can be easily tailored to specific applications. Our results demonstrate the superior performance of the proposed approach on data from simulation and synthetic data, including in real-world applications like morphologies clustering. [Display omitted] [ABSTRACT FROM AUTHOR]

Wastewater treatment plant (WWTP) discharges of microcontaminants negatively impact freshwater streams, underscoring the need for accurately mapping wastewater-derived contamination in water bodies across Europe (EU). In this study, we present a fast and open-source code for a microcontaminants fate and transport (MFT) model, capable of simulating contamination at a high resolution (15 arc second scale) across the EU. The model was developed using the best publicly available hydrological (HydroSHEDS) and sanitation (UWWTD) datasets and was rigorously calibrated, with a goodness of fit of 77.5% as measured by the R 2. Importantly, the model demonstrated the ability to predict wastewater-derived contamination in water bodies, making it a valuable tool for planning the upgrade of WWTPs and improving the ecological condition of freshwater streams in the EU. • A model is developed to simulate wastewater-based contamination in rivers. • Large-scale free hydrologic and sanitation layers are used. • The model is calibrated obtaining an R2 of 77.5%. • The model is open-source and has fast runtimes. [ABSTRACT FROM AUTHOR]

In smart manufacturing, an increasing amount of heterogeneous data inputs need to be integrated into a holistic traceability model. One key challenge is the coherent mapping of dynamic and event-based sensor data for real-time tracking purposes with static data, which are mostly needed for retrospective tracing purposes. In this research, we develop a graph-based traceability model, which integrates static and dynamic data sub-models in one connected traceability graph. For an automotive use case, the novel traceability model is implemented and tested using a hybrid architecture based on the graph database Neo4j and the event engine Apache Kafka. [ABSTRACT FROM AUTHOR]

The problem of automatic analysis and processing of electroencephalogram (EEG) signals has been an attractive but a quite difficult research subject with many potential clinical applications. For this problem, one major goal is to find and locate EEG state change that is from a normal state to an abnormal state during monitoring period, which remains challenging due to the high complexity of EEG signals. This study proposes a dynamic modeling method based on the combination of dynamic graph analysis and hierarchical decomposition (HD). The method can take into account the difference of contribution of individual frequency components in the EEG by construction of a family of hierarchical graph models (HGMs) from frequency components decomposed by HD. Adaptive input weighting (AIW) fusion is subsequently adopted to utilize and assemble these differences from all constructed graphs to form a comprehensive indicator that can describe the behavioral characteristics of EEG signals. Finally, a common null hypothesis testing is allowed for producing a decision making. Experiments results, our method can successfully detect all seizures without false alarm outperforming representable methods, demonstrated the availability and great potentials of the proposed method in real scenarios. This work is an extension of a recent result (Lu et al., 2020), which introduces an automated system that can detect changes in a given EEG time series. • Innovative Hierarchical Graph Modeling This approach combines dynamic graph analysis with hierarchical decomposition to address the complexity and diverse frequency components of EEG signals. • Adaptive Input Weighting Fusion Utilizing adaptive input weighting (AIW), this approach integrates the diverse contributions of individual frequency components into a comprehensive indicator. • Superior Performance and Validation Through experiments on two public datasets, this approach is proven to be a powerful clinical tool for early disease diagnosis and monitoring. [ABSTRACT FROM AUTHOR]

The quality of polyester yarn is mainly affected by its dyeing uniformity. As a result, textile manufacturers need to inspect the dyeing uniformity of polyester yarn. The existing inspection methods rely on the dyeing process which can be viewed as a way to extract the discriminative information of the polyester yarns with different dyeing uniformities. However, the dyeing process is extremely time-consuming and laborious. It is very expensive to obtain the dyeing uniformities of polyester yarns, due to the inefficient dyeing process. To address this problem, an efficient and effective inspection method based on near-infrared imaging spectroscopy and semi-supervised learning is introduced in this paper. This introduced method can obtain the dyeing uniformity of polyester yarn without the dyeing process, which greatly boosts the efficiency of inspection. In the developed method, to acquire the representative information without the dyeing process, near-infrared spectra are applied to represent the molecular structure differences of different polyester yarns. Considering that the dyeing uniformities are expensive to obtain, a novel semi-supervised learning algorithm, termed consensus local graph based on multiple kernel learning (CLGMKL), is proposed in the introduced method to evaluate the dyeing uniformity of polyester yarn. Since redundant wavebands and noise exist in the collected spectra, the relationship among spectra in CLGMKL is learned based on the extracted low-rank matrices which contain the discriminative information of base kernels. Besides, CLGMKL incorporates the label information of spectra. Finally, extensive experiments on the collected spectra of polyester yarns demonstrate the superiority of the proposed method. • We use NIR spectra to represent the differences among PET yarns. • A novel semi-supervised learning algorithm is proposed to handle the spectra. • We can directly obtain the dyeing properties without the dyeing process. [ABSTRACT FROM AUTHOR]

Manoeuvrable, responsive satellite constellations that respond to real time events could provide data on demand for time-critical tasks, such as disaster monitoring and relief efforts. The authors demonstrate the feasibility of such a system by expanding on a fully analytical method for designing responsive spacecraft manoeuvres using low thrust propulsion. These manoeuvres are perceived as a graph that enables efficient exploration and optimised selection of favourable routes that achieve mission goals while highlighting resilience and redundancy in the mission's execution. A case study is presented that considers four satellites required to provide flyovers of two targets, with an associated graph of 10,839 possible manoeuvres. Investigation of the graph highlights that a good, but not minimum time, solution can allow the system to perform well, while also providing greater resilience to changes in mission priorities and errors in execution. This analytical approach enables operators to trade-off between a loss of time by using only one satellite versus the disruption of moving multiple satellites for a potentially faster response. The impact of varying mission capabilities, such as using fewer satellites, smaller swath width or less propellant, can be evaluated by reducing the graph without recalculating manoeuvre options. [ABSTRACT FROM AUTHOR]

Nearest neighbor search (NNS) is one of the main concerns in data stream applications since similarity queries can be used in multiple scenarios. Online NNS is usually performed on a sliding window by lazily scanning every element currently stored in the window. This paper proposes Sliding Window-based Incremental Nearest Neighbors (SWINN), a graph-based online search index algorithm for speeding up NNS in potentially never-ending and dynamic data stream tasks. Our proposal broadens the application of online NNS-based solutions, as even moderately large data buffers become impractical to handle when a naive NNS strategy is selected. SWINN enables efficient handling of large data buffers by using an incremental strategy to build and update a search graph supporting any distance metric. Vertices can be added and removed from the search graph. To keep the graph reliable for search queries, lightweight graph maintenance routines are run. According to experimental results, SWINN is significantly faster than performing a naive complete scan of the data buffer while keeping competitive search recall values. We also apply SWINN to online classification and regression tasks and show that our proposal is effective against popular online machine learning algorithms. • We introduce SWINN, a graph-based nearest neighbor search index for sliding windows. • SWINN works with any distance metric and is robust against concept drift. • SWINN is significantly faster than a linear scan of the data buffer. • Our proposal delivers competitive search recalls compared to the traditional strategy. • SWINN is successfully applied to classification and regression case studies. [ABSTRACT FROM AUTHOR]

Arabic handwritten words segmentation is considered a challenging step in Arabic optical character recognition. In this work, we propose a convolutional neural network-based segmentation of handwritten Arabic words. After generating candidate segmentation points, we applied a graph theory-based technique to divide the word into sub-words. Our approach reached a 96% correct sub-word segmentation rate. Additionally, we develop a Segmentation Hypothesis Graph (SHG) to generate candidate characters for each sub-word. We employ both a manually designed convolutional neural network and a transfer learning-based CNN using the pretrained AlexNet model. The accuracy, 96.97%, of the pretrained model outperformed the manually and state of the art models trained on the same dataset. Finally, we used recognition confidence to validate the segmentation points. Hence, our approach correctly segments 88% of the words into characters and successfully resolves 100% of the overlapping cases. [ABSTRACT FROM AUTHOR]

In 1975, P. Erdős proposed the problem of determining the maximum number f (n) of edges in a graph with n vertices in which any two cycles are of different lengths. The sequence (c 1 , c 2 , ... , c n) is the cycle length distribution of a graph G with n vertices, where c i is the number of cycles of length i in G. Let f (a 1 , a 2 , ... , a n) denote the maximum possible number of edges in a graph which satisfies c i ≤ a i , where a i is a nonnegative integer. In 1991, Shi posed the problem of determining f (a 1 , a 2 , ... , a n) which extended the problem due to Erdős. It is clear that f (n) = f (1 , 1 , ... , 1). Let g (n , m) = f (a 1 , a 2 , ... , a n) , where a i = 1 if i ∕ m is an integer, and a i = 0 otherwise. It is clear that f (n) = g (n , 1). We prove that lim inf n → ∞ f (n) − n n ≥ 2 + 40 99 , which is better than the previous bounds 2 (Shi (1988)), and 2 + 7654 19071 (Lai (2017)). We show that lim inf n → ∞ g (n , m) − n n m > 2. 444 , for all even integers m. We make the following conjecture: lim inf n → ∞ f (n) − n n > 2. 444. [ABSTRACT FROM AUTHOR]

We introduce the set G SSP of all simple graphs G with the property that each symmetric matrix corresponding to a graph G ∈ G SSP has the strong spectral property. We find several families of graphs in G SSP and, in particular, characterise the trees in G SSP. [ABSTRACT FROM AUTHOR]

Computer-aided engineering (CAE) applications generate vast quantities of heterogeneous data. Domain experts often fail to explore and analyze these data, because they are not integrated across different applications. Existing data management solutions are rather tailored to scientific applications. In our approach, we tackle this issue by combining a data lake solution with graph-based metadata management. This provides a holistic view of all CAE data and of the data-generating applications in one interconnected structure. Based on a prototypical implementation, we discuss how this eases the task of domain experts to explore and extract data for further analyses. [ABSTRACT FROM AUTHOR]

Intrusion detection has been the subject of numerous studies in industry and academia. Still, Cybersecurity analysts always want greater precision and global threat analysis to secure their systems in cyberspace. To improve the intrusion detection system, the visualization of the security events in the form of graphs and diagrams is significant to improve the accuracy of alerts. In this paper, we propose an approach of an IDS based on cloud computing, big data technique and, using a machine learning graph algorithm which can detect in real-time different attacks as early as possible. We use the MAWILab intrusion detection dataset. We choose Microsoft Azure as a unified cloud environment to load our dataset on Azure blob storage. We implement the k2 algorithm, which is a graphical machine learning algorithm to classify attacks. Our system showed a great performance due to the graphical machine learning algorithm and Apache Spark structured streaming engine. [ABSTRACT FROM AUTHOR]

We compute the treewidth of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. Our main technique is constructing strict brambles of large orders. We discuss connections to divisorial graph theory coming from tropical geometry, and use our results to compute the divisorial gonality of these graphs. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n (G) vertices and e (G) edges. The elementary cyclic number c (G) of G is defined as c (G) = e (G) − n (G) + ω (G) , where ω (G) is the number of connected components of G. The nullity of G , denoted by η (G) , is the multiplicity of the eigenvalue zero of the adjacency matrix of G. A graph is leaf-free if it has no pendent vertices. In Ma et al. (2016) proved that if G is leaf-free and each component of G contains at least two vertices, then η (G) ≤ 2 c (G) , the equality is attained if and only if G is the union of disjoint cycles, where each cycle has length a multiple of 4. In this paper, we completely characterize all leaf-free graphs with nullity one less than the above upper bound, i.e., η (G) = 2 c (G) − 1. [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]

Let G be a simple graph of order n and also let ρ (G) be the spectral radius of the graph G. In this paper, we prove that ρ (G) ≤ ρ 2 (G − v k) + 2 d k − 1 for any non-isolated vertex v k of degree d k in G , thus confirming a conjecture posed by Guo et al. (2019) [3]. Moreover, we characterized all connected graphs for which this bound is attained. [ABSTRACT FROM AUTHOR]

A set of vertices W in a graph G is called resolving if for any two distinct x , y ∈ V (G) , there is v ∈ W such that d G (v , x) ≠ d G (v , y) , where d G (u , v) denotes the length of a shortest path between u and v in the graph G. The metric dimension md (G) of G is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether md (G) ⩽ k , is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by k was proved to be W [ 2 ] -hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether md (G) ⩽ n − k , where n is the order of G. We prove that the dual parameterization admits (a) a kernel with at most 6 (k + 1) vertices and (b) a randomized algorithm of runtime O ⁎ (4 k + o (k)). Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number v c (G) of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by v c (G) + k , unless NP ⊆ coNP/poly. Our reduction also gives evidence for non-existence of polynomial Turing kernels. We also prove that Metric Dimension parameterized by bandwidth or cutwidth does not admit a polynomial kernel, unless NP ⊆ coNP/poly. Finally, using Eppstein's results (2015) we show that Metric Dimension parameterized by max-leaf number does admit a polynomial kernel. [ABSTRACT FROM AUTHOR]

A link stream is a sequence of pairs of the form (t , { u , v }) , where t ∈ N represents a time instant and u ≠ v. Given an integer γ , the γ -edge between vertices u and v , starting at time t , is the set of temporally consecutive edges defined by { (t ′ , { u , v }) | t ′ ∈ 〚 t , t + γ − 1 〛 }. We introduce the notion of temporal matching of a link stream to be an independent γ -edge set belonging to the link stream. We show that the problem of computing a temporal matching of maximum size is NP-hard as soon as γ > 1. We depict a kernelization algorithm parameterized by the solution size for the problem. As a byproduct we also give a 2-approximation algorithm. Both our 2-approximation and kernelization algorithms are implemented and confronted to link streams collected from real world graph data. We observe that finding temporal matchings is a sensitive question when mining our data from such a perspective as: managing peer-working when any pair of peers X and Y are to collaborate over a period of one month, at an average rate of at least two email exchanges every week. We furthermore design a link stream generating process by mimicking the behavior of a random moving group of particles under natural simulation, and confront our algorithms to these generated instances of link streams. All the implementations are open source. [ABSTRACT FROM AUTHOR]

The structure of an undirected graph is completely determined by a symmetric matrix: its adjacency matrix with respect to an ordering of its vertices; and that matrix can be used to define an integral quadratic form. The main purpose of this paper is to raise this question: "What can quadratic forms tell us about graphs?" As an initial answer, the theory of quadratic forms will be applied to the graph isomorphism problem. The essential definitions and facts from the theory of quadratic forms will be sketched without proof. [ABSTRACT FROM AUTHOR]

This work introduces a general concept of center for graphs, built on the model of the characteristic set ([12,14]) of trees. We define it as the set of cycles in a specific directed graph associated with the original graph G , and we let it depend on a function μ. In the case of trees we consider particular instances of μ given as weights of rooted subtrees, thus retrieving the characteristic set and, interestingly, the eccentricity-center. We investigate when the center of a graph G is simple – i.e., consisting of a unique cycle – and quasi-simple – i.e., inducing a connected subgraph of G. In particular, we prove that the center of a caterpillar tree associated with the so-called combinatorial Perron parameter ρ c (studied in [2] and [3]) is always simple. We also make use of a discrete version of concavity to generate examples of simple and quasi-simple centers for graphs. [ABSTRACT FROM AUTHOR]

We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix and the Laplacian distance matrix of a graph. We also obtain a classification of the graphs with at most one trivial distance ideal. [ABSTRACT FROM AUTHOR]

The nullity η (G) of G is the multiplicity of 0 as an eigenvalue of A (G). In this paper, we completely solve the following conjecture proposed by Zhou, Wong and Sun in [Linear Algebra and its Applications, 555 (2018) 314-320]: Let G be a connected graph of order n with nullity η (G) and the maximum degree Δ ≥ 2. Then η (G) ≤ (Δ − 2) n + 2 Δ − 1 , the equality holds if and only if G ≅ C n (n ≡ 0 (m o d 4)) or G ≅ K Δ , Δ. [ABSTRACT FROM AUTHOR]

Given a graph G and an integer k , two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins if at the end of the game all the vertices of G are colored. The game chromatic number χ g (G) is the minimum k for which the first player has a winning strategy. In this study, we prove that the game chromatic number of the splitting graphs of the path P n and cycle C n for n ≥ 5 is 4. We also answer a question posed by Xuding Zhu in [12] for the splitting graphs of path P n and n -cycle for all n ≥ 3 [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n vertices v 1 , ... , v n and m edges. Let A (G) = [ a i j ] n × n be the adjacency matrix of G , that is a i j = 1 if v i and v j are adjacent, and a i j = 0 , otherwise. Let λ 1 , ... , λ n be the eigenvalues of A (G). The energy of G , denoted by E (G) , is defined as | λ 1 | + ⋯ + | λ n |. It is well known that 2 m ≤ E (G) ≤ 2 m n. In this paper we improve the latter lower bound and prove that if all eigenvalues of G are non-zero, then E (G) ≥ 2 | λ 1 λ n | | λ 1 | + | λ n | 2 m n , where | λ 1 | ≥ ⋯ ≥ | λ n |. We show that the equality holds if and only if G = n 2 K 2 or G = p K β + 1 ⋃ q T β + 1 , where p and q are some non-negative integers and β ≥ 2 and T β + 1 is the graph K β + 1 , β + 1 ∖ M , where M is a perfect matching of K β + 1 , β + 1. Finally by studying the | λ 1 λ n | | λ 1 | + | λ n | of graphs, we find some lower bounds for the energy of graphs. In particular, we obtain that if G has no eigenvalue in the interval (− 1 , 1) , then E (G) ≥ 2 m 2 n − 2 n , and the equality holds if and only if G is the complete graph K n. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥ ... ≥ λ n its eigenvalues. The Estrada index of G is defined as E E (G) = ∑ i = 1 n e λ i . In this work, we use an increasing sequence converging to the λ 1 to obtain an increasing sequence of lower bounds for E E (G). In addition, we generalize this succession for the Estrada index of an arbitrary symmetric nonnegative matrix. [ABSTRACT FROM AUTHOR]

This paper generalizes and unifies the existing spectral bounds on the k -independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than k. The previous bounds known in the literature follow as a corollary of the main results in this work. We show that for most cases our bounds outperform the previous known bounds. Some infinite families of graphs where the bounds are tight are also presented. Finally, as a byproduct, we derive some spectral lower bounds for the diameter of a graph. [ABSTRACT FROM AUTHOR]

Let M be a partitioned matrix and B be its equitable quotient matrix. In this paper, we prove the inclusion of their spectra σ (B) ⊆ σ (M) , and the equality of their spectral radii ρ (B) = ρ (M) if M is nonnegative. Moreover, σ (M) is shown to be completely determined by σ (B) if M is from a class of special matrices. Then we apply these results to obtain the spectra of some well-known matrices associated with graphs or digraphs, and then determine the extremal values of different spectral radii associated with connected graphs or strongly connected digraphs with given vertex connectivity. [ABSTRACT FROM AUTHOR]

Let G = (V (G) , E (G)) be a simple graph with vertex set V (G) and edge set E (G) , c (G) = | E (G) | − | V (G) | + θ (G) be the dimension of cycle spaces of G with θ (G) the number of connected components of G , m (G) be the matching number of G , and η (G) be the nullity of G. It was shown in Wang and Wong (2014) that the nullity η (G) of G is bounded by an upper bound and a lower bound as | V (G) | − 2 m (G) − c (G) ≤ η (G) ≤ | V (G) | − 2 m (G) + 2 c (G). Graphs with nullity attaining the upper bound and the lower bound have been characterized by Song et al. (2015) and Wang (2016), respectively. In this paper, we prove that there is no graph with nullity η (G) = | V (G) | − 2 m (G) + 2 c (G) − 1. For fixed c (G) , infinitely many connected graphs with nullity | V (G) | − 2 m (G) + 2 c (G) − k , (0 ≤ k ≤ 3 c (G) , k ≠ 1) are also constructed. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with n vertices and m edges. The spectral radius ρ (G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of ρ (G) which improves some known bounds: If (k − 2) (k − 3) 2 ≤ m − n ≤ k (k − 3) 2 , where k (3 ≤ k ≤ n) is an integer, then ρ (G) ≤ 2 m − n − k + 5 2 + 2 m − 2 n + 9 4 . The equality holds if and only if G is a complete graph K n or K 4 − e , where K 4 − e is the graph obtained from K 4 by deleting some edge e. [ABSTRACT FROM AUTHOR]

In this article, we obtain a sufficient condition related to toughness τ (G) for a graph to be all fractional (a , b , k) -critical. We prove that if τ (G) ≥ (b 2 − 1) + a k a for some nonnegative integers a , b , k , then G is all fractional (a , b , k) -critical. Our result improves the known results in Liu and Zhang (2008) and Liu and Cai (2009). [ABSTRACT FROM AUTHOR]

A well-known conjecture of Grünbaum (1970) and Nash-Williams (1971) asserts that every 4-connected toroidal graph has a Hamiltonian cycle. Related to this conjecture, Kawarabayashi and Ozeki (2011) proved two results on a 2-factor and a 3-factor. In this paper, motivated by these results, we give several sufficient conditions for a graph embedded in a surface to have an [ a , b ] -factor. We also show that several conditions are best possible. [ABSTRACT FROM AUTHOR]

We associate an adequate graph to any pair (V , W) where V is a graded module over a graded linear space W , in such a way that allows us to study the inner algebraic structure of (V , W). In particular, the homogeneous indecomposability, the homogeneous semisimplicity and the homogeneous simplicity of (V , W) are characterized in terms of this graph. Some applications to the theory of non-associative algebras and modules over algebras are also provided. [ABSTRACT FROM AUTHOR]

Let V be a linear space of arbitrary dimension and over an arbitrary base field F , endowed with a bilinear map f : V × V → V. A basis B = { v i } i ∈ I of V is an f -basis if for any i , j ∈ I we have that f (v i , v j) ∈ F v k for some k ∈ I. We associate to any triplet (V , f , B) an adequate graph (V , E). By arguing on this graph we show that V decomposes as a direct sum of strongly f -invariant linear subspaces, each one being associated to one connected component of (V , E). Also the B -semisimplicity and the B -simplicity of V are characterized in terms of the associated graph. [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 An independent 2-clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside. In this paper, we study the complexity of finding an independent 2-clique of maximum size in several graph classes and we compare it with the complexity of maximum independent set. We prove that this problem is NP-hard on apex graphs, APX-hard on line graphs, not n 1 / 2 − ϵ -approximable on bipartite graphs and not n 1 − ϵ -approximable on split graphs, while it is polynomial-time solvable on graphs of bounded degree and their complements, graphs of bounded treewidth, planar graphs, (C 3 , C 6) -free graphs, threshold graphs, interval graphs and cographs. [ABSTRACT FROM AUTHOR]

Abstract Let G be a graph of order n. Also let λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n be the eigenvalues of graph G. In this paper, we present the following upper bound on the sum of the k (≤ n) largest eigenvalues of G in terms of the order n and negative inertia θ (the number of negative eigenvalues): ∑ i = 1 k λ i ≤ n 2 (θ + 1) (θ + θ (k θ + k − 1) ) with equality holding if and only if G ≅ n K 1 or G ≅ p K ‾ t (n = p t , p ≥ 2) with k = 1 (where p K ‾ t is the complete p -partite graph of p t vertices with all partition sets having size t). Moreover, we obtain a sharp upper bound on the energy of bipartite graphs with given order n and rank r. We also propose an open problem as follows: Characterize all connected d -regular bipartite graphs of order n with 5 distinct eigenvalues and rank r = 2 n 2 (4 d − n) 2 , where d > n 4. Finally we give a partial solution to this problem. [ABSTRACT FROM AUTHOR]

Abstract A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌊ (n + 1) ∕ 2 ⌋. Gallai's Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai's Conjecture for graphs with maximum degree at most 5, and Botler et al. (2017) verified it for graphs with treewidth at most 3. In this paper, we verify Gallai's Conjecture for triangle-free planar graphs. [ABSTRACT FROM AUTHOR]

Abstract Let G = (V , E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S , and every vertex in V − S is adjacent to a vertex in V − S. The total restrained domination number of G , denoted γ t r (G) , is the smallest cardinality of a total restrained dominating set of G. In Koh et al. (2013), it is proved that if G is a graph of order n ≥ 4 and δ (G) ≥ 2 , then γ t r (G) ≤ n − n 4 3. It is further conjectured that this bound can be improved to γ t r (G) ≤ n − θ (n). In this paper we show that if G is a graph with no C 3 components and δ (G) ≥ 2 , then γ t r (G) ≤ n − n 2 . [ABSTRACT FROM AUTHOR]

Abstract Mahlmann and Schindelhauer (2005) defined a Markov chain which they called k -Flipper, and showed that it is irreducible on the set of all connected regular graphs of a given degree (at least 3). We study the 1-Flipper chain, which we call the flip chain, and prove that the flip chain converges rapidly to the uniform distribution over connected 2 r -regular graphs with n vertices, where n ≥ 8 and r = r (n) ≥ 2. Formally, we prove that the distribution of the flip chain will be within ε of uniform in total variation distance after poly (n , r , log (ε − 1)) steps. This polynomial upper bound on the mixing time is given explicitly, and improves markedly on a previous bound given by Feder et al. (2006). We achieve this improvement by using a direct two-stage canonical path construction, which we define in a general setting. This work has applications to decentralised networks based on random regular connected graphs of even degree, as a self-stabilising protocol in which nodes spontaneously perform random flips in order to repair the network. [ABSTRACT FROM AUTHOR]

Abstract The fixing number of a graph Γ is the minimum number of labeled vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of Γ. The fixing set of a finite group G is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to G. Previously, authors have studied the fixing sets of both abelian groups and symmetric groups. In this article, we determine the fixing set of the dihedral group. [ABSTRACT FROM AUTHOR]

Highlights • Optimal configuration requires minimum number of CFD calculations. • The set covering approaches ensures 100% coverage for a given length of the optimisation mesh. • CFD simulations are not mandatory for optimisation of gas detectors. • High performance and low computational cost. • 3D visualisation of the final configuration of gas detectors. Abstract An approach to optimise the number and location of gas detectors was developed based on the colour pattern of the graph of the set covering problem (SCP). The optimisation problem is combined to Computational Fluid Dynamics (CFD) data and the discrete optimisation problem is solved using Balas algorithm in the framework of a developed Fortran code. Every computational cell is regarded as a node of a graph where the links are the common boards shared by the neighbouring cells. The graph is read into the code via a dedicated algorithm for generation of the set of constrains. The characteristic length that determines the distance among the nodes of the graph is obtained from the set of CFD simulations or any stablished criterion. This work is based on the discovery of the colour pattern observed in the adjacency matrix that represents the set of constrains of the optimisation formulation and also on the limited set of gas dispersion CFD simulations to ensure full coverage of the area. Two cases (covering problem and p-median problem) are used in the validation of the developed code. Results are compared with the results obtained using CPLEX. Additional tests are considered ranging from simple 2D cases with 4 nodes up to 75 nodes. Two real engineering cases are considered and the efficiency of the method is discussed based on the volume of the cloud detected and the time to detection. Results show an efficient method for optimisation of gas detectors able to detect flammable clouds as small as a few cubic meters within 10 seconds. [ABSTRACT FROM AUTHOR]