Let G=(V, E) be a GRAPH with p vertices and q edges. An acyclic GRAPHOIDAL cover of G is a collection y of paths in G which are internally-disjoint and cover each edge of the GRAPH exactly once. Let f:V® {1, 2, …, P} be a bijective LABELing of the vertices of G. Let ­Gf be the directed GRAPH obtained by orienting the edgesuv of G from u to v provided f(u)<f(v). If the set yf of all maximal directed paths in ­Gf, with directions ignored, is an acyclic GRAPHOIDAL cover of G, then f is called a GRAPHOIDAL LABELing of G and G is called a LABEL GRAPHOIDAL GRAPH and  hi=min {½yf½: f is a GRAPHOIDAL LABELing of G} is called the LABEL GRAPHOIDAL covering number of G. In this paper we characterize GRAPHs for which (i) hi=q-m, where m is the number of vertices of degree 2 and (ii) hi=q. Also, we determine the value of LABEL GRAPHOIDAL covering number for unicyclic GRAPHs.

This paper explores GRAPH embedding techniques for effectively analyzing large, heterogeneous GRAPHs with complex and noisy patterns. GRAPHs represent data through nodes (entities) and edges (relationships), and when dealing with large-scale data, effective search methods are crucial. GRAPH embedding helps evaluate node significance and transforms data into latent space representations. It also addresses challenges like handling multi-LABEL data in heterogeneous networks, where nodes may have multiple LABELs describing complex concepts. Traditional methods struggle with such multi-LABEL scenarios and fail to capture LABEL dependencies. The paper introduces a GRAPH Neural Network (GCN)-based node embedding method, which extends traditional neural networks to GRAPH data. GCNs allow the extraction of local features from nodes and their neighbors, making them useful for heterogeneous networks. By integrating LABEL information into the embedding process, the method improves relationships between LABELs. The proposed approach transforms neighboring LABELs into continuous vectors, structured into a matrix for learning. This enhances the overall network embedding. The method outperforms previous techniques, demonstrating improved performance on real-world datasets, such as a 2.4% improvement on the IMDB dataset and 9.3% on the DBLP dataset. The paper discusses GRAPH embedding techniques in the first section and explores the potential of multi-LABEL embedding in non-uniform GRAPHs, suggesting future research directions in the final section. The article's code link on GitHub can also be found at the following: https://github.com/frshkara/EGSA.

A GRAPHOIDAL cover of a GRAPH (not necessarily finite) is a collection of paths (not necessarily finite, not necessarily open) satisfying the following axioms: (GC-1) Every vertex of is an internal vertex of at most one path in , and (GC-2) every edge of is in exactly one path in . The pair is called a GRAPHOIDALly covered GRAPH and the paths in are called the -edges of . In a GRAPHOIDALly covered GRAPH , two distinct vertices and are -adjacent if they are the ends of an open -edge. A GRAPHOIDALly covered GRAPH in which no two distinct vertices are -adjacent is called -independent and the GRAPHOIDAL cover is called a totally disconnecting GRAPHOIDAL cover of . Further, a GRAPH possessing a totally disconnecting GRAPHOIDAL cover is called a GRAPHOIDALly independent GRAPH. The aim of this paper is to establish complete characterization of GRAPHOIDALly independent infinite cactus.

A difference LABELing of a GRAPH G is an injective function f: V (G) → N ∪ {0} together with the weight function f∗ on E(G) given by f∗(uv) = |f(u)-f(v)| for every edge uv in G. The collection of subGRAPHs induced by the edges of the same weight is a decomposition of G and is called the common weight decomposition of G induced by f. Let ϒf denote the collection of all the paths taken from each member of the common weight decomposition induced by f. A difference LABELing f of G is said to be a GRAPHOIDAL difference LABELing if ϒf is an acyclic GRAPHOIDAL decomposition of G. This paper initiates a study on this concepts.

Multi-LABEL classification aims at assigning more than one LABEL to each instance. Many real-world multi-LABEL classification tasks are high dimensional, leading to reduced performance of traditional classifiers. Feature selection is a common approach to tackle this issue by choosing prominent features. Multi-LABEL feature selection is an NP-hard approach, and so far, some swarm intelligence-based strategies and have been proposed to find a near optimal solution within a reasonable time. In this paper, a hybrid intelligence algorithm based on the binary algorithm of particle swarm optimization and a novel local search strategy has been proposed to select a set of prominent features. To this aim, features are divided into two categories based on the extension rate and the relationship between the output and the local search strategy to increase the convergence speed. The first group features have more similarity to class and less similarity to other features, and the second is redundant and less relevant features. Accordingly, a local operator is added to the particle swarm optimization algorithm to reduce redundant features and keep relevant ones among each solution. The aim of this operator leads to enhance the convergence speed of the proposed algorithm compared to other algorithms presented in this field. Evaluation of the proposed solution and the proposed statistical test shows that the proposed approach improves different classification criteria of multi-LABEL classification and outperforms other methods in most cases. Also in cases where achieving higher accuracy is more important than time, it is more appropriate to use this method.

Classification is a crucial process in data mining, data science, machine learning, and the applications of natural language processing. Classification methods distinguish the correlation between the data and the output classes. In single-LABEL classification (SLC), each input sample is associated with only one class LABEL. In certain real-world applications, data instances may be assigned to more than one class. The type of classification which is required in such applications is known as multi-LABEL classification (MLC). In MLC, each sample of data is associated with a set of LABELs. Due to the presence of multiple class LABELs, the SLC learning process is not applicable to MLC tasks. Many solutions to the multi-LABEL classification problem have been proposed, including BR, FS-DR, and LLSF. But, these methods are not as accurate as they could be. In this paper, a new multi-LABEL classification method is proposed based on GRAPH representation. A feature selection technique and the Q-learning method are employed to increase the accuracy of the proposed algorithm. The proposed multi-LABEL classification algorithm is applied to various standard multi-LABEL datasets. The results are compared with state-of-the-art algorithms based on the well-known performance evaluation metrics. Experimental results demonstrated the effectiveness of the proposed model and its superiority over the other methods.

On many occasions, the evaluation of a phenomenon based on a single feature could not solely be resulted in comprehensive and accurate results. Moreover, even if we have several features, we don’ t know in advance, which feature offers a better description of the phenomenon. Thus, selecting the best features and especially their combination could lead to better results. An affinity GRAPH is a tool that can describe the relationship between the samples. In this paper, we proposed a GRAPH-based sample-based ranking method that sorts the GRAPHs based on six proposed parameters. The sorting is performed such that the GRAPHs at the top of the list have better performance compared to the GRAPHs at the bottom. Furthermore, we propose a fusion method to merge the information of various features and improve the accuracy of LABEL propagation. Moreover, a method is proposed for parameter optimizations and the ultimate decision fusion. The experimental results indicate that the proposed scheme, apart from correctly ranking the GRAPHs according to their accuracy, in the fusion step, increases the accuracy compared to the use of a single feature.

The independence GRAPH Ind(G) of a GRAPH G is the GRAPH with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence GRAPH of Cartesian product of d copies of complete GRAPHs Kq, which is known as the Hamming GRAPH H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming GRAPH H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.

In this paper, we find the star chromatic number of central GRAPH of complete bipartite GRAPH and corona GRAPH of complete GRAPH with path and cycle.