It was shown by Kitaev and Pyatkin, in , that if a graph is representable by , then one can assume that is uniform, that is, it contains the same number of copies of each letter. If the number of copies of each letter in is , we say that is -uniform. For example, the graph to the left in Figure 1 can be represented by the 2-uniform word 12312434 (in this word every pair of letters alternate, except 1 and 4, and 2 and 4), while the graph to the right, the Petersen graph,
We will consider triangulations of a polyomino. Note that no triangulation is 2-colorable – at least three colors are needed to color properly a triangulation, while four colors is always enough to colour any triangulation since we deal with planar graphs and it is well-known that such graphs are 4-colorable . Not all triangulations of a polyomino are 3-colorable – for example, see Figure 1 for non-3-colorable triangulations (which are straightforward to check to require four colors, and also to be the only such triangulations, up to rotations, of a 3 × 3 grid graph). The main result of this paper is the following theorem.
It would be interesting to apply this methodology to wordgraphs generated with a language model, although this way of generating the graphs would not fit exactly the theoretical model. If a language model is used to generate the graphs, then their lex- ical confusion could be reduced, so better results could be achieved. Other interesting task in which this methodology could help is in performing SLU experiments on a combination of the output of some different ASR engines. All these interesting appli- cations constitute a line of our future work.
Related work. The notion of directed word-representable graphs was in- troduced in  to obtain asymptotic bounds on the free spectrum of the widely-studied Perkins semigroup, which has played central role in semi- group theory since 1960, particularly as a source of examples and coun- terexamples. In , numerous properties of word-representable graphs were derived and several types of word-representable and non-word-representable graphs pinpointed. Some open questions from  were resolved recently in , including the representability of the Petersen graph.
be one of the four graphs presented in Figure 5 to the right of the leftmost graph. However, in each of the four cases, we have a vertex labeled by * that would require colour 4 contradicting the fact that v is supposed to be the only vertex coloured by 4. This completes our considerations of eight of subcases in the situation S 1 out of 36. The remaining subcases are to be
We compare the proposed method against several word representation methods in Table 2. All methods in Table 2 use 200 dimensional vectors to represent a word. A baseline method is created that shows the level of performance we can reach if we represent each word u as a vector of patterns l in which u occurs. First, we create a co-occurrence matrix between words u and patterns l, and use Singular Value De- composition (SVD) to create 200 dimensional projections for the words. Because patterns represent contexts in which words appear in the corpus, this baseline can be seen as a version of the Latent Semantic Analysis (LSA), that has been widely used to represent words and documents in infor- mation retrieval. Moreover, SVD reduces the data sparseness in raw co-occurrences. We create three versions of this base- line denoted by SVD+LEX, SVD+POS, and SVD+DEP corresponding to relational graphs created using respec- tively LEX, POS, and DEP patterns. CBOW (Mikolov et al. 2013b), skip-gram (Mikolov et al. 2013c), and GloVe (Pen-
The triangular tiling graph T ∞ (i.e., the two-dimensional triangular grid) is the Archimedean tiling 3 6 introduced in  and . By a triangular grid graph G in this paper we mean a graph obtained from T ∞ as follows. Specify a number of triangles, called cells, in T ∞ . The edges of G are then all the edges surrounding the specified cells, while the vertices of G are the endpoints of the edges (defined by intersecting lines in T ∞ ). We say that the specified cells, along with any other cell whose all edges are from G, belong to G. Any triangular grid graph is 3-colorable, and thus it is word-representable . We consider non-3-colorable graphs obtained from triangular grid graphs by applying the operation of face subdivision which is defined in the sequel.
Our proof is organized as follows. In Subsection 4.1 we will provide all six minimum non-word-representable graphs that can appear in triangulations of GCCGs with three sectors (see Figure 4.11) and give an explicit proof that one of these graphs is non-word-representable. Then, in Subsection 4.2, we will give an inductive argument showing that avoidance of the six graphs in Figure 4.11 is a sufficient condition for a GCCG with three sectors to be word-representable. Note that the graphs in Figure 4.11 were obtained by an exhaustive computer search on graphs on up to eight vertices. However, our argument in Subsection 4.2 will show that no other non-word-representable induced subgraphs can be found among all triangulations of GCCGs with three sectors.
The concept of fuzzy graphs was initiated by Kaufmann , based on Zadeh’s fuzzy relations. Later, another elaborated definition of fuzzy graph with fuzzy vertex and fuzzy edges was introduced by Rosenfeld  and obtaining analogs of several graph theoretical concepts such as paths, cycles and connectedness etc, he developed the structure of fuzzy graphs. Some remarks on fuzzy graphs were given by Bhattacharya . Fuzzy linegraphs were studied in  by Mordeson. Nair and Cheng  defined the concept of a fuzzy clique consistent with the definition of fuzzy cycles in fuzzy graphs. Intuitionistic fuzzy graphs with vertex sets and edge sets as IFS were introduced by Akram and Davvaz . Sahoo and Pal [19, 20] introduced some new concepts of intuitionistic fuzzy graphs. Naz et al. [3, 16, 17] put forward many new concepts concerning the extended structures of fuzzy graphs. Kandasamy et al.  put forward the notion of neutrosophic graphs. Neutrosophic graphs, particularly SVNGs [2, 4, 8, 9, 14, 15] have attracted significant interest from researchers in recent years. In literature, the study of SVNLGs and SVNCs is still blank. To fill this vacancy, we shall focus on the study of SVNLGs and SVNCs, in this paper.
 O.Ore, Theory of graphs, Amer.Math. Soc.Colloq. Publ., 38, Providence, (1962).  T.W.Hayness, S.T.Hedetniemi and P.J.Slater, Fundmentals of Domination in graphs, Marcel,Dekker,Inc, Newyork(1998)  E.Sampathkumar, The global domination number of a graph J.Math Phys.Sci.23 (1989) 377-385.
duced using the same algorithms with the same hyper-parameters. We evaluate Con- neau et al. (2018) on pairs of embeddings induced with different hyper-parameters in §4.4. While keeping hyper-parameters fixed is always possible, it is of practical interest to know whether the unsupervised methods work on any set of pre-trained word embeddings. We also investigate the sensitivity of unsuper- vised BDI to the dimensionality of the monolin- gual word embeddings in §4.5. The motivation for this is that dimensionality reduction will alter the geometric shape and remove characteristics of the embedding graphs that are important for alignment; but on the other hand, lower dimensionality intro- duces regularization, which will make the graphs more similar. Finally, in §4.6, we investigate the impact of different types of query test words on performance, including how performance varies across part-of-speech word classes and on shared vocabulary items.
ﬁrst characterization (partition into complete subgraphs) was given by Krausz . Since this is a survey on generalizations of linegraphs, we will not describe linegraphs and their properties in any detail here. Instead, we refer the interested reader to a somewhat older but still an excellent survey on linegraphs and line digraphs by Hemminger and Beineke . A more recent book by Prisner  describes many interesting generalizations of linegraphs. For general graph theoretic concepts and terminology not deﬁned here, please see [9, 16].
Students discover the relationship between algebraic expressions and verbal expressions. They apply their knowledge of basic operations to expressions that include variables. They use the order of operations to solve open sentence equations and inequalities containing a vari- able. Students learn to recognize and use the properties of identity and equality, and the Dis- tributive, Commutative, and Associative Prop- erties. They use these properties to simplify expressions and evaluate equations. Students use tables and coordinates to draw graphs of
V ∪ , and two vertices are adjacent if and only if they are adjacent or incident in G . It is introduced by Behzad & Chartrand . Several properties of total graphs are investigated in the literature (see [1-4,6,7,15,20]). The total graph H = T ( G ) of G is shown in Figure 1.
The following lemma has been proved in  for zero-divisor graphs. Exactly the same proof will work for maximal graph, i.e., for Γ(R). Lemma 3.2. Let R be a finite ring. Then L(Γ(R)) is Eulerian if and only if deg(v) is even for all v ∈ V (Γ(R)) or deg(v ) is odd for all v ∈ V (Γ(R)).
Now introduce a point on any nonboundary line in the plane embedding of G but not on the boundary line because if we introduce a point on the boundary line then one point and a line is additional and inner points remain same. The additional line formed in G is again a nonboundary line and by Lemma 2, it corresponds to an inner point in L(G). Thus L(G) is (n+1)-minimally nonouterplanar. This completes the proof of the theorem.
In this section, we determine all of those trees having planar 3-linegraphs. First, we state a well-known characterization of planar graphs. A graph H is a subdivision of a graph G if H can be obtained from G by inserting vertices of degree 2 into some, all or none of the edges of G. Clearly, a subdivision H of a graph G is planar if and only if G is planar. The graphs K 5 and K 3 , 3 and their subdivisions play a pivotal role in the study of planar graphs.