A graph G = (V, E) is **word**-**representable** if there exists a **word** w over the alphabet V such that letters x and y alternate in w if and only if (x, y) ∈ E for each x 6= y . The set of **word**-**representable** **graphs** generalizes several important and well-studied graph families, such as circle **graphs**, compa- rability **graphs**, 3-colorable **graphs**, **graphs** of vertex degree at most 3, etc. By answering an open question from [9], in the present paper we show that not all **graphs** of vertex degree at most 4 are **word**-**representable**. Combining this result with some previously known facts, we derive that the number of n -vertex **word**-**representable** **graphs** is 2 n 3 2 +o(n

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we raise some concerns about Conjecture 7, while confirming it for **graphs** on at most 9 vertices. In Section 3 we present a complementary computational approach using constraint programming, enabling us count connected non-**word**-**representable** **graphs**. In particular, in Section 3 we report that using 3 years of CPU time, we found out that 64.65% of all connected **graphs** on 11 vertices are non-**word**-**representable**. Another important corollary of our **results** in Section 3 is the correction of the published result [19, 20] on the number of connected non- **word**-**representable** **graphs** on 9 vertices (see Table 2). In Section 4 we introduce the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation, and show that 3-semi-transitively orientable **graphs** are not necessarily semi-transitively orientable. Finally, in Section 5 we suggest a few directions for further research and experimentation.

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A graph G = V E , is **representable** if there exists a **word** W over the alphabet V such that letters x and y alternate in W if and only if x y , is in E for each x not equal to . The motivation to study **representable** **graphs** came from algebra, but this subject is interesting from graph theoretical, com- puter science, and combinatorics on words points of view. In this paper, we prove that for greater than 3, the line graph of an -wheel is non-**representable**. This not only provides a **new** construction of non-repre- sentable **graphs**, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-**representable** graph. Moreover, we show that for greater than 4, the line graph of the com- plete graph is also non-**representable**. We then use these facts to prove that given a graph which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of -times is guaranteed to be non-**representable** for greater than 3.

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In this paper we extend the **results** of Akrobotu, Kitaev and Mas´ arov´ a [1] to the case of grid-covered cylinder **graphs**, which is a cyclic version of rect- angular grid **graphs**; see Subsection 2.2 for definitions. It turns out that in this case, some of the **graphs** in question with chromatic number 4 are actu- ally **word**-**representable**; for example, see the underlying graph in Figure 3.7. Still, assuming that there are at least four sectors in a grid-covered cylinder graph, **word**-**representable** triangulations of such **graphs** are characterized by avoidance of W 5 and W 7 as induced subgraphs. On the other hand, we

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The paper is organized as follows. In the rest of the section, we give more details about **word**-**representable** **graphs**. In Section 2, we introduce rigorously the notion of a k-11- **representable** graph and provide a number of general **results** on these **graphs**. In particular, we show that a (k − 1)-11-**representable** graph is necessarily k-11-**representable** (see Theo- rem 2.2). In Section 3, we study the class of 1-11-**representable** **graphs**. These studies are extended in Section 4, where we 1-11-represent all non-**word**-**representable** **graphs** on at most 7 vertices. In Section 5 we prove that any graph is 2-11-**representable**. Finally, in Section 6, we state a number of open problems on k-11-**representable** **graphs**.

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Recently, a number of (fundamental) **results** on **word**-**representable** **graphs** were obtained in the literature; for example, see [1], [3], [5], [7], [9], [11], and [12]. In particular, Halld´ orsson et al. [7] have shown that a graph is **word**- **representable** if and only if it admits a semi-transitive orientation (to be de- fined in Section 2), which, among other important corollaries, implies that all 3-colorable **graphs** are **word**-**representable**. The theory of **word**-**representable** **graphs** is the main subject of the upcoming book [8].

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The task of **word** sense disambiguation (WSD) can be regarded as one of the most important tasks for natural language processing applications including semantic interpretation of texts, semantic web applications, paraphrasing and summarization. One issue of current **word** sense disambiguation methods is that the most successful techniques are supervised, which means that annotated corpora should be available to train the systems. However, this kind of data is heavy to produce and cannot be created for each **new** domain to be disambiguated. This indicates that more efforts should be put on unsupervised **word** sense disambiguation techniques. Furthermore, one vital issue that should generally be solved for this kind of systems is the choice of an adequate context. Usually, this context is defined as a window of words or sentences around the **word** to be disambiguated. The question raised by this paper is whether defining this context using syntactic and logical features can be beneficial to WSD. This paper briefly presents a natural language processing pipeline that outputs logical representations from texts and disambiguates the logical representations using various WSD algorithms. The paper also presents different context definitions that are used for WSD. Preliminary **results** show that logical and syntactic features can be of interest to WSD. The main contribution of this paper is the use of syntactic and semantic information for WSD in an unsupervised manner. The paper is organized as follows: First, section 2 explains the pipeline that creates logical representations and presents the various WSD algorithms and the contexts used in this study. Section 3 presents experiments that are conducted over a small corpus and shows preliminary **results**. It also describes the **results** of our system on the Senseval English lexical Sample Task before drawing a conclusion.

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we raise some concerns about Conjecture 7, while confirming it for **graphs** on at most 9 vertices. In Section 3 we present a complementary computational approach using constraint programming, enabling us count connected non-**word**-**representable** **graphs**. In particular, in Section 3 we report that using 3 years of CPU time, we found out that 64.65% of all connected **graphs** on 11 vertices are non-**word**-**representable**. Another important corollary of our **results** in Section 3 is the correction of the published result [19, 20] on the number of connected non- **word**-**representable** **graphs** on 9 vertices (see Table 2). In Section 4 we introduce the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation, and show that 3-semi-transitively orientable **graphs** are not necessarily semi-transitively orientable. Finally, in Section 5 we suggest a few directions for further research and experimentation.

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In this paper, we present WordGraph2Vec, a **word** embedding algorithm with semantic enhancement. The algorithm makes use of both linear and graph input in order to strengthen the semantic relations between words. Our experimental **results** show that the proposed embedding did not achieve the best **results** on analogy and classification tasks but was stable across the datasets and in most cases was ranked at the second place in terms of docu- ment classification and analogy tests accuracy. In future work, further settings of WordGraph2Vec can be explored, such as additional **word** graph configurations and a larger radius R for the **new** target words, which should yield target words that are not close to the context **word** in the original text. In addition, the proposed graph-based ap- proach to **word** embedding can be evaluated on other NLP tasks in multiple languages.

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Organization of the paper. The paper is organized as follows. In Sec- tion 2, we give definitions of objects of interest and review some of the known **results**. In Section 3, we give a characterization of **word**-**representable** **graphs** in terms of orientations and discuss some important corollaries of this fact. In Section 4, we examine the representation number, and show that it is always at most 2n − 4, but can be as much as n/2. We explore, in Section 5, which classes of **graphs** are **word**-**representable**, and show, in particular, that 3-colorable **graphs** are such **graphs**, but numerous other properties are inde- pendent from the property of being **word**-**representable**. Finally, we conclude with two open problems in Section 6.

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While multilingual **word** vectors have been evaluated with respect to intrinsic parameters such as embedding dimensionality, empirical work on another aspect appears to be lacking: the second language involved. For example, it might be the case that projecting two languages with very different lexical semantic associations in a joint embedding space inherently deteriorates monolingual embeddings as measured by performance on an intrinsic monolingual semantic evaluation task, relative to a setting in which the two languages have very similar lexical semantic associations. To illustrate, the classical Latin **word** vir is sometimes translated in English as both ‘man’ and ‘warrior’, suggesting a semantic connotation, in Latin, that is putatively lacking in English. Hence, projecting English and Latin in a joint semantic space may invoke semantic relations that are misleading for an English evaluation task. Alternatively, it may be argued that heterogeneity in semantics between the two languages involved is beneficial for monolingual evaluation tasks in the same way that uncorrelatedness in classifiers helps in combining them.

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and complete bipartite fuzzy **graphs**. The bounds is obtained for the vertex domination number of fuzzy **graphs**. Also the relationship between M -strong arcs and α-strong is obtained. In fuzzy **graphs**, monotone decreasing property and monotone increasing property is introduced. We prove the vizing’s conjecture is monotone decreasing fuzzy graph property for vertex domination. we prove also the Grarier-Khelladi’s conjecture is monotone decreasing fuzzy graph property for it. We obtain Nordhaus-Gaddum (NG) type **results** for these parameters. The relationship between several classes of operations on fuzzy **graphs** with the vertex domination number of them is studied.

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Unless mentioned otherwise for terminology and notation the reader may refer Buckley and Harary [2] and Chartrand and Lensiak [3], **new** ones will be introduced as and when found necessary. In this paper we consider simple undirected **graphs** without multiple edges and self loops. The order p is the number of vertices in G and size q is the number of edges in G. The distance d(u, v) between u and v is the length of a shortest path joining u and v. If there exists no path between u and v then we define d(u, v) = ∞. The eccentricity e(u) of u is the distance to a vertex farthest from u. If d(u, v) = e(u)(v 6= u), we say that v is an eccentric vertex of u. The radius rad(G) is the minimum eccentricity of the vertices, where as the diameter diam(G) is the maximum eccentricity. A vertex v is a central vertex if e(v) = rad(G), and the center C(G) is the set of all central vertices. A graph G is self-centered if rad(G) = diam(G). The join of two **graphs** G 1 and G 2 , defined by Zykov [8], is denoted G 1 + G 2 and consists of

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[2]M.Akram, Bipolar fuzzy **graphs**, Information sciences,DOI 10.1016/j.ins 2011.07.037,2011. [3] A.Nagoorgani,K.Radha, On regular fuzzy **graphs**, journal of physical sciences,Vol.12,33-44,2008. [4] A.Nagoorgani and j.Malarvizhi properties of 𝜇-complement of a fuzzy graph, international journal of algorithms, Computing and Mathematics, Vol.2,No.3,73-83, 2009.

All **graphs** G = ( V ( G ) , E ( G )) in this paper are finite, connected and undirected. For any undefined nota- tions and terminology we follow [3]. If the vertices or edges or both of the graph are assigned valued subject to certain conditions it is known as graph labeling. A dynamic survey on graph labeling is regularly updated by Gallian [4]. Labeled **graphs** have variety of applications in graph theory, particularly for missile guidance code, design good radar type codes and convolution codes with optimal autocorrelation properties. Labeled **graphs** plays vital role in the study of X-ray crystallography, communication network and to determine opti- mal circuit layouts. A detailed study on variety of applications on graph labeling is carried out in Bloom and Golomb [1].

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Let LE,F be the space of all bounded linear operators from E into F and BE* the unit ball of E*, the dual of E The completion of the injective tensor product of E and F is denoted by E F[r]

Residuated algebras and their equational theories have been investigated on their own right and also in connection with substructural logics. The reason for the latter is that the algebraizations of substructural logics like relevance logic [AB75, ABD92] and the Lambek calculus (LC) [La58] yield residuated algebras. Indeed, for these logics, the Lindenbaum–Tarski alge- bras are residuated algebras and sound relational semantics can be provided using families of binary relations, i.e., **representable** residuated algebras. These connections are explained in detail in [Mik??] and the references therein. In particular, we show in [Mik??] completeness of an expansion of LC with meet w.r.t. binary relational semantics. This completeness re- sult states that that derivability in LC augmented with derivation rules for meet coincides with semantic validity, i.e., completeness is stated in its weak form and does not capture general semantic consequence. The proof uses cut-elimination. In algebraic terms this result means that the equational theories of abstract (related to the syntactic calculus) and **representable** (related to binary semantics) algebras coincide. In other words, the free abstract algebra is **representable**.

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the Entity-**Word** graph, even Unseen-entities can explicitly be connected with other entities. Our method encodes the graph with Graph Convolu- tional Networks (GCNs) (Kipf and Welling, 2017) to learn entity representations considering the global features of the entire graph. GCNs simplify the convolutional operations on the graph, and learn node representations based on their neigh- borhood information. GCNs are utilized for sev- eral NLP tasks (Zhang et al., 2018; De Cao et al., 2019). By encoding the Entity-**Word** graph with GCNs, not only the descriptions information but also information of the related entities is propa- gated to the Unseen-entities through words. We expect that the entity representations learned via our Entity-**Word** graph can contribute to the im- provement in the performance of the KGC.

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Another graph-based method is presented in (Dorow and Widdows, 2003). They extract only noun neighbours that appear in conjunctions or dis- junctions with the target **word**. Additionally, they extract second-order co-occurrences. Nouns are rep- resented as vertices, while edges between vertices are drawn, if their associated nouns co-occur in con- junctions or disjunctions more than a given num- ber of times. This co-occurrence frequency is also used to weight the edges. The resulting graph is then pruned by removing the target **word** and ver- tices with a low degree. Finally, the MCL algorithm (Dongen, 2000) is used to cluster the graph and pro- duce a set of clusters (senses) each one consisting of a set of contextually related words.

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