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We consider the problem of finding an accurate representation of neuron shapes, extracting sub-cellular features, and classifying neurons based on neuron shapes. In neuroscience research, the skeleton representation is often used as a compact and abstract representation of neuron shapes. However, existing methods are limited to getting and analyzing curve skeletons which can only be applied for tubular shapes. This paper presents a 3D neuron morphology analysis method for more general and complex neuron shapes. First, we introduce the concept of skeleton mesh to represent general neuron shapes and propose a novel method for computing mesh representations from 3D surface point clouds. A skeleton graph is then obtained from skeleton mesh and is used to extract sub-cellular features. Finally, an unsupervised learning method is used to embed the skeleton graph for neuron classification. Extensive experiment results are provided and demonstrate the robustness of our method to analyze neuron morphology.

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a finite, combinatorial graph. We define a notion of curvature on the vertex set V via the inverse of the resistance distance matrix. We prove that this notion of curvature has a number of desirable properties. Graphs with curvature bounded from below by K > 0 have diameter bounded from above. The Laplacian L = D − A satisfies a Lichnerowicz estimate, there is a spectral gap λ 2 ≥ 2 K. We obtain matching two-sided bounds on the maximal commute time between any two vertices in terms of | E | ⋅ | V | − 1 ⋅ K − 1 . Moreover, we derive quantitative rates for the mixing time of the corresponding Markov chain and prove a general equilibrium result. [ABSTRACT FROM AUTHOR]

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d , d ≤ 4 , is at most c d n where c 3 = 1 and c 4 = 9. Whether such a constant c d exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, Γ ≇ K 3 , 3 , has a regular orbit, that is, an orbit of 〈 g 〉 of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C. [ABSTRACT FROM AUTHOR]

Early detection and diagnosis of Autism Spectrum Disorder (ASD) can significantly improve the quality of life for affected individuals. Identifying ASD based on brain functional connectivity (FC) poses a challenge due to the high heterogeneity of subjects’ fMRI data in different sites. Meanwhile, deep learning algorithms show efficacy in ASD identification but lack interpretability. In this paper, a novel approach for ASD recognition is proposed based on graph attention networks. Specifically, we treat the region of interest (ROI) of the subjects as node, conduct wavelet decomposition of the BOLD signal in each ROI, extract wavelet features, and utilize them along with the mean and variance of the BOLD signal as node features, and the optimized FC matrix as the adjacency matrix, respectively. We then employ the self-attention mechanism to capture long-range dependencies among features. To enhance interpretability, the node-selection pooling layers are designed to determine the importance of ROI for prediction. The proposed framework are applied to fMRI data of children (younger than 12  years old) from the Autism Brain Imaging Data Exchange datasets. Promising results demonstrate superior performance compared to recent similar studies. The obtained ROI detection results exhibit high correspondence with previous studies and offer good interpretability. [ABSTRACT FROM AUTHOR]

Background: The development of Brain-Computer Interface (BCI) technology has brought tremendous potential to various fields. In recent years, prominent research has focused on enhancing the accuracy of BCI decoding algorithms by effectively utilizing meaningful features extracted from electroencephalographic (EEG) signals. Objective: This paper proposes a method for extracting brain functional network features based on directed transfer function (DTF) and graph theory. The method incorporates the extracted brain network features with common spatial pattern (CSP) to enhance the performance of motor imagery (MI) classification task. Methods: The signals from each electrode of the EEG, utilizing a total of 32 channels, are used as input signals for the network nodes. In this study, 26 healthy participants were recruited to provide EEG data. The brain functional network is constructed in Alpha and Beta bands using the DTF method. The node degree (ND), clustering coefficient (CC), and global efficiency (GE) of the brain functional network are obtained using graph theory. The DTF network features and graph theory are combined with the traditional signal processing method, the CSP algorithm. The redundant network features are filtered out using the Lasso method, and finally, the fused features are classified using a support vector machine (SVM), culminating in a novel approach we have termed CDGL. Results: For Beta frequency band, with 8 electrodes, the proposed CDGL method achieved an accuracy of 89.13%, a sensitivity of 90.15%, and a specificity of 88.10%, which are 14.10, 16.69, and 11.50% percentage higher than the traditional CSP method (75.03, 73.46, and 76.60%), respectively. Furthermore, the results obtained with 8 channels were superior to those with 4 channels (82.31, 83.35, and 81.74%), and the result for the Beta frequency band were better than those for the Alpha frequency band (87.42, 87.48, and 87.36%). Similar results were also obtained on two public datasets, where the CDGL algorithm's performance was found to be optimal. Conclusion: The feature fusion of DTF network and graph theory features enhanced CSP algorithm's performance in MI task classification. Increasing the number of channels allows for more EEG signal feature information, enhancing the model's sensitivity and discriminative ability toward specific activities in brain regions. It should be noted that the functional brain network features in the Beta band exhibit superior performance improvement for the algorithm compared to those in the Alpha band. [ABSTRACT FROM AUTHOR]

For a simple connected graph G of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥⋯ ≥ μn−1 > μn = 0, the Laplacian resolvent energy of G is defined as RL(G) =∑i=1n 1 n+1−μi. In this paper, we provide an improved lower bound for RL(G) through majorization. Considering our lower bound, we also derive some lower bounds for RL(G) when the graph G possesses tree structure. [ABSTRACT FROM AUTHOR]

Let Qd ${Q}_{d}$ be the hypercube of dimension d $d$ and let H $H$ and K $K$ be subsets of the vertex set V(Qd) $V({Q}_{d})$, called configurations in Qd ${Q}_{d}$. We say that K $K$ is an exact copy of H $H$ if there is an automorphism of Qd ${Q}_{d}$ which sends H $H$ onto K $K$. Let n≥d $n\ge d$ be an integer, let H $H$ be a configuration in Qd ${Q}_{d}$ and let S $S$ be a configuration in Qn ${Q}_{n}$. We let λ(H,d,n) $\lambda (H,d,n)$ be the maximum, over all configurations S $S$ in Qn ${Q}_{n}$, of the fraction of sub‐d $d$‐cubes R $R$ of Qn ${Q}_{n}$ in which S∩R $S\cap R$ is an exact copy of H $H$, and we define the d $d$‐cube density λ(H,d) $\lambda (H,d)$ of H $H$ to be the limit as n $n$ goes to infinity of λ(H,d,n) $\lambda (H,d,n)$. We determine λ(H,d) $\lambda (H,d)$ for several configurations in Q3 ${Q}_{3}$ and Q4 ${Q}_{4}$ as well as for an infinite family of configurations. There are strong connections with the inducibility of graphs. [ABSTRACT FROM AUTHOR]

Let G be a connected graph on n vertices with adjacency matrix A G. Associated to G is a polynomial d G (x 1 , ... , x n) of degree n in n variables, obtained as the determinant of the matrix M G (x 1 , ... , x n) , where M G = Diag (x 1 , ... , x n) − A G. We investigate in this article the set V d G (r) of non-negative values taken by this polynomial when x 1 , ... , x n ≥ r ≥ 1. We show that V d G (1) = Z ≥ 0. We show that for a large class of graphs one also has V d G (2) = Z ≥ 0. When V d G (2) ≠ Z ≥ 0 , we show that for many graphs V d G (2) is dense in Z ≥ 0. We give numerical evidence that in many cases, the complement of V d G (2) in Z ≥ 0 might in fact be finite. As a byproduct of our results, we show that every graph can be endowed with an arithmetical structure whose associated group is trivial. [ABSTRACT FROM AUTHOR]

In recent years, the drone swarm has experienced remarkable growth, finding applications across diverse domains such as agricultural surveying, disaster rescue and logistics delivery. However, the rapid expansion of drone swarm usage underscores the necessity for innovative approaches in the field. Traditional algorithms face challenges in adapting to complex tasks, environmental modeling and computational complexity, highlighting the need for more advanced solutions like multi-agent deep reinforcement learning to enhance efficiency and robustness in drone swarm. Our proposed approach tackles this challenge by embracing temporal and spatial. In terms of the temporal, the proposed approach builds upon historical data, it enhances the predictive capabilities regarding future behaviors. In the spatial, the proposed approach leverage graph theory to model the swarm's features, while attention mechanisms strengthen the relationships between individual drones. The proposed approach addresses the unique characteristics of drone swarms by incorporating temporal dependencies, spatial structures and attention mechanisms. Extensive experiments validate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with vertex set V(G) = {v1, v2, ..., vn} and edge set E(G), and let d v i be the degree of the vertex vi. Let D(G) be the distance matrix and let Tr(G) be the diagonal matrix of the vertex transmissions of G. The generalized distance matrix of G is defined as Dα(G) = αTr(G) + (1 − α)D(G), where 0 ⩽ α ⩽ 1. Let λ1(Dα(G)) ⩾ λ2(Dα(G)) ⩾ ... ⩾ λn(Dα(G)) be the generalized distance eigenvalues of G, and let k be an integer with 1 ⩽ k ⩽ n. We denote by Sk(Dα(G)) = λ1(Dα(G)) + λ2(Dα(G)) +... + λk (Dα(G)) the sum of the k largest generalized distance eigenvalues. The generalized distance spread of a graph G is defined as DαS(G) = λ1(Dα(G)) − λn(Dα(G)). We obtain some bounds on Sk((Dα(G))) and DαS(G) of graph G, respectively. [ABSTRACT FROM AUTHOR]

In recent years, the Internet has had a main and important contribution to human life and the amount of data on the World Wide Web such as books, movies, videos and, etc. increase rapidly. Recommender systems allow users to quickly access items that are closer to their interests. One of the most popular and easiest models of recommender systems is the Collaborative Filtering (CF) model, which uses the items ratings given by users. The important challenge of CF is robust against the attacks which manipulated by fake users to reduce the efficiency of the system. Therefore, the impact of attacks on the item recommendations will increase and fake items will be easily recommended to users. The purpose of this paper is to design a robust CF recommender system, T&TRS, Time and Trust Recommender System, against user attacks. Our proposed system improves the performance of users clustering for detecting the fake users based on a novel community detection algorithm that is introduced in this paper. Our proposed system calculates the reliably value for all items ratings and tags them as suspicious or correct. T&TRS considered the rating time and implicit and explicit trust among users for constructing the weighted user-user network and detects communities as the nearest neighbors of the users to predict unknown items ratings. After detecting the suspect users and items using a novel community detection method, our proposed system removes them from rating matrix and predict the rating of unobserved items and generate the Top@k items according to the user interests. We inject the random and average attacks into the Epinion data set and evaluate our proposed systems based on Precision, Recall, F1, MAE, RMSE, and RC measures before and after attacks. The experimental results indicated that the precision of items recommendations increase after attack detection and show the effectiveness of T&TRS in comparison to the two base K-means methods such as KMCF-U, KMCF-I and graph-based methods such as TRACCF, and TOTAR. [ABSTRACT FROM AUTHOR]

Wireless Sensor Networks (WSNs) comprise interconnected wireless nodes that receive and transmit data across applications and platforms. This paper addressed the problem of link failures in WSNs that potentially could lead to the loss of data packets while still in transit. This was achieved through the use of network coding which is known to address capacity bottleneck problems in WSNs. In particular, a technique called Ant Agent-Assisted Network Coding (AAANC) is proposed that employs the ant colony optimization technique in addition to network coding operations. The main aim of AAANC is to facilitate the successful delivery and decoding of coded data packets in the presence of link failures. AAANC employs a packet route selection technique that is inspired by the social behavior of natural ants. For natural ants, a strong pheromone trail along a path indicates a promising route to a food source, and this is analogous to a reliable communication link for routing data packets in this paper. Through simulations, AAANC was compared to diagonal pseudorandom network coding (DNC) and triangular pseudorandom network coding (TNC), and it proved to have a superior performance in terms of packet delivery ratio and number of decoded packets. Significant performance gain can be achieved if AAANC algorithm is made to dynamically adapt the ant colony and network coding parameters in response to traffic changes. [ABSTRACT FROM AUTHOR]

The use of graph theory in the fields of chemistry, pharmacy, communication, maps, and aeronautics is significant. In order to study the properties of chemical compounds, the molecules of those compounds are modeled as graphs. Boiling point, enthalpy, π-electron energy, and molecular weight are a few examples of physical properties that are related to the geometric structure of the compound. Recently, the modified symmetric division deg (mSDD(G), in short) index is defined as the total of all adjacent vertices in pairs μv of the term ... . The purpose of this article is to demonstrate the usefulness of mSDD(G) index through the resolution of an interdisciplinary problem describing the structure of benzenoid hydrocarbons. With the help of linear regression models, we have studied the physicochemical properties of benzenoid hydrocarbons. Strong correlations were obtained, and the bounds for the same index were subsequently established. [ABSTRACT FROM AUTHOR]

A spanning subgraph F of G is called a path factor if every component of F is a path of order at least 2. Let k ≥ 2 be an integer. A P≥k-factor of G means a path factor in which every component has at least k vertices. A graph G is called a P≥k-factor avoidable graph if for any e ∈ E(G), G has a P≥k-factor avoiding e. A graph G is called a (P≥k, n)-factor critical avoidable graph if for any W ⊆ V (G) with |W| = n, G − W is a P≥k-factor avoidable graph. In other words, G is (P≥k, n)-factor critical avoidable if for any W ⊆ V (G) with |W| = n and any e ∈ E(G − W), G − W − e admits a P≥k-factor. In this article, we verify that (i) an (n + r + 2)-connected graph G is (P≥2, n)-factor critical avoidable if I(G)>(n+r+2)/(2(r+2)) I (G) > n + r + 2 2 (r + 2) $I{(G)}>\frac{n+r+2}{2{(r+2)}}$ ; (ii) an (n + r + 2)-connected graph G is (P≥3, n)-factor critical avoidable if t(G)>(n+r+2)/(2(r+2)) t (G) > n + r + 2 2 (r + 2) $t{(G)}>\frac{n+r+2}{2{(r+2)}}$ ; (iii) an (n + r + 2)-connected graph G is (P≥3, n)-factor critical avoidable if I(G)>(n+3(r+2))/(2(r+2)) I (G) > n + 3 (r + 2) 2 (r + 2) $I{(G)}>\frac{n+3{(r+2)}}{2{(r+2)}}$ ; where n and r are two nonnegative integers. [ABSTRACT FROM AUTHOR]

It is known that a strict contraction on complete metric spaces with graphs possesses a fixed point. In the present paper we show that this property holds for mappings on complete metric spaces with graphs which are merely contractive. We also obtain some stability results. [ABSTRACT FROM AUTHOR]

We provide answers to open questions from Banerjee and Gopaulsingh (Bull Pol Acad Sci Math 71: 1–21, 2023) about the relationship between the Erdős–Dushnik–Miller theorem (EDM ) and certain weaker forms of the Axiom of Choice (AC ), and we properly strengthen some results from Banerjee and Gopaulsingh (2023). We also settle a part of an open question of Lajos Soukup (stated in Banerjee and Gopaulsingh (2023) [Question 6.1]) about the relationship between the following two order-theoretic principles, which [as shown in Banerjee and Gopaulsingh (2023)] are weaker than EDM : (a) "Every partially ordered set such that all of its antichains are finite and all of its chains are countable is countable" (this is known as Kurepa's theorem), and (b) "Every partially ordered set such that all of its antichains are countable and all of its chains are finite is countable". In particular, we prove that (b) does not imply (a) in ZF (i.e., Zermelo–Fraenkel set theory without AC ). Moreover, with respect to (b), we answer an open question from Banerjee and Gopaulsingh (2023) about its relationship with the following weak choice form: "Every set is either well orderable or has an amorphous subset"; in particular, we show that (b) follows from, but does not imply, the latter weak choice principle in ZFA (i.e., Zermelo–Fraenkel set theory with atoms). [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex set V (G) = { v 1 , v 2 , ... , v n } . Recall that the reciprocal distance signless Laplacian matrix of G is defined to be R Q (G) = R T (G) + R D (G) , where RD(G) is the reciprocal distance matrix, and R T i is the reciprocal distance degree of vertex v i for i = 1 , 2 , ... , n , R T (G) = diag (R T 1 , R T 2 , ... , R T n) . Denote by μ 1 (R Q (G)) and μ n (R Q (G)) the largest eigenvalue and the least eigenvalue of RQ(G), respectively. The reciprocal distance signless Laplacian spread of G is defined as S RQ (G) = μ 1 (R Q (G)) - μ n (R Q (G)) . In this paper, we obtain some bounds on reciprocal distance signless Laplacian spread of a graph. [ABSTRACT FROM AUTHOR]

A graph G is called fractional [a, b]-covered if for every e ∈ E (G) , G has a fractional [a, b]-factor containing e. A graph G is called fractional (a, b, k)-critical covered if after deleting any k vertices of G, the remaining graph of G is fractional [a, b]-covered. In this paper, we pose a Fan-type condition for a graph being fractional (a, b, k)-critical covered, which is an improvement of Zhou, Xu and Sun's previous result [S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional (a, b, k)-critical covered graphs, Information Processing Letters 152(2019)105838]. Furthermore, we claim that the main result in this paper is best possible in some sense. [ABSTRACT FROM AUTHOR]

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The primary objective of this study was to propose a functional discrete mathematical model for analyzing folklore fairy tales. Within this model, characters are denoted as vertices, and explicit instances of communication – both verbal and non-verbal – within the text are depicted as edges. Upon examining a corpus of Eastern Slavic fairy tales in comparison to Chukchi fairy tales, unforeseen outcomes emerged. Notably, the constructed models seem to evade establishing certain connections between characters. Consequently, instances where the interactions among fairy tale characters would result in a non-planar graph structure are notably absent. To put it differently, the models refrain from incorporating sub-graphs delineated by the Kuratowski theorem governing planar graphs, specifically the minimal non-planar graphs Κ5 and Κ3,3. Remarkably, even in more extensive texts featuring a larger cast of characters, connections that would yield a non-planar graph pattern are consistently avoided. This leads to the formulation of a hypothesis positing that traditional folk tales adhere to a "planar narrative" design – an identifiable narrative variant characterized by inherent limitations in complexity. This design, in turn, appears deeply entrenched within the societal framework of the cultures that produced these folk narratives. [ABSTRACT FROM AUTHOR]

Data have become valuable assets for enterprises. Data governance aims to manage and reuse data assets, facilitating enterprise management and enabling product innovations. A data lineage graph (DLG) is an abstracted collection of data assets and their data lineages in data governance. Analyzing DLGs can provide rich data insights for data governance. However, the progress of data governance technologies is hindered by the shortage of available open datasets for DLGs. This paper introduces an open dataset of DLGs, including the DLG model, the dataset construction process, and applied areas. This real-world dataset is sourced from Huawei Cloud Computing Technology Company Limited, which contains 18 DLGs with three types of data assets and two types of relations. To the best of our knowledge, this dataset is the first open dataset of DLGs for data governance. This dataset can also support the development of other application areas, such as graph analytics and visualization.

A methodology is proposed for modifying computer ontologies (CO) for electronic courses (EC) in the field of information and communication technologies (ICT) for universities, schools, extracurricular institutions, as well as for the professional retraining of specialists. The methodology includes the modification of CO by representing the formal ontograph of CO in the form of a graph and using techniques for working with the graph to find optimal paths on the graph using applied software (SW). A genetic algorithm (GA) is involved in the search for the optimal CO. This will lead to the division of the ontograph into branches and the ability to calculate the best trajectory in a certain sense through the EC educational material, taking into account the syllabus. An example is considered for the ICT course syllabus in terms of a specific topic covering the design and use of databases. It is concluded that for the full implementation of this methodology, a tool is needed that automates this procedure for developing EC and/or electronic textbooks. An algorithm and a prototype of software tools are also proposed, integrating machine methods of working with CO and graphs.

The objective of this study is to investigate the final exam scheduling process at the Ho Chi Minh City Open University and develop an automated exam scheduling application. Our primary objectives are to prevent students from having conflicting exam schedules and to ensure that no student has to take more than two exams on the same day. This research focuses on applying graph coloring algorithms to the problem of automatic exam scheduling. Our research findings indicate that the graph coloring algorithm is highly effective for automated exam scheduling. This study has the potential to expand and support the development of an automatic exam scheduling and management system, in line with our overall goals. We conduct the experiments on the practical data at HCMCOU and obtain promising results.

The main aim of the paper is to give the crossing number of the join product G* + Dn for the graph G* isomorphic to 4-regular graph on six vertices except for two distinct edges with no common vertex such that two remaining vertices are still adjacent, and where Dn consists of n isolated vertices. The proofs are done with possibility of an existence of a separating cycle in some particular drawing of the investigated graph G* and also with the help of well-known exact values for crossing numbers of join products of two subgraphs Hk of G* with discrete graphs. [ABSTRACT FROM AUTHOR]

This article is concerned with the classification of Schur covering groups of the elementary abelian group of order 2 n , up to isomorphism. We consider those covering groups possessing a generating set of n elements having only two distinct squares. We show that such groups may be represented by 2-vertex-colored and 2-edge-colored graphs of order n. We show that in most cases, the isomorphism type of the group is determined by that of the 2-colored graph, and we analyze the exceptions. [ABSTRACT FROM AUTHOR]

In recent years, developments in the fields of computer vision and artificial intelligence have created new opportunities for studying sports performance. With advancements in computer vision and artificial intelligence, it is now possible to use massive volumes of video data to get deeper insights into sports dynamics, especially in precision-based video sports like table tennis. To develop skills, a thorough examination of player movements is required. With the development of vision-based human posture recognition, computers can function like humans and derive intelligent judgments from outside data. This paper presents a novel method that uses graph convolutional neural networks (GCNNs) to detect and identify improper postures and movements in table tennis players. In addition to conventional methods, the proposed method dissects the human skeleton into finely detailed head, trunk, and leg features. Deep-level features that offer a more comprehensive understanding of athlete movements are extracted after feeding these features into the network. The softmax classifier combines these features to produce the final recognition result. The effectiveness of this approach has been evaluated through extensive experimentation. The GCN model performs remarkably well, with accuracy rates of 86.4% on the NTU-RGB + D dataset and 79.1% on the COCO dataset. This accomplishment is significant because the model consistently yields accurate detection results, even in complex and occlusion-filled scenes. In comparison tests, the proposed model performs better than GraphSAGE, GAT, ChebNet, GIN, and GC-LSTM in identifying relevant body part movements in table tennis players while ignoring irrelevant ones. According to proposed experiments, this enhances the recognition of improper postures in most table tennis actions. [ABSTRACT FROM AUTHOR]

Hierarchical, descriptive, and faceted methods of constructing knowledge classifiers and also the classification of knowledge using folksnomies are described. Mathematical models to formalize the listed methods of constructing classifiers are presented. A general description of the classifiers of the Russian Science Citation Index, Code of State Categories Scientific and Technical Information, and Universal Decimal Classification is given. The Chinese experience of classifying scientific publications and the American experience of classifying patent documents are analyzed. An alternative possibility for classifying scientific publications using clustering with bibliometric indicators and using keywords is indicated. A review of the main methods and means of comparing knowledge classifiers using qualifiers and expert competencies is carried out. A new approach to the compilation of knowledge classifiers based on their reduction to oriented trees and the construction of homomorphisms between these graphs is proposed. [ABSTRACT FROM AUTHOR]