edges. This type of argument, and its obvious analogue for decreasing graph properties, maximally decouple geometric and graph theoretic considerations. For the lower tail of the **clique** **number**, our results, Theorem 2 (i) and (ii), leave open the possibility that such an argument could yield tight bounds for threshold dimensions. For the remaining properties we consider, our results rule this out – the dimensional thresholds cannot be explained by edge density alone.

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In this paper weintroduce the concept maximum and minimumnumber of edges in the graph to colour k vertices by m colours, and introduce some results related to this concept. In section four we used colouring of families of disjoint sets technique, to colour vertices of the graph depend on the **clique** **number**, and we introduce some results of vertex’scolouring (trivial colouring and nontrivialcolouring) depend on **clique** **number**. In section five weprove a result about **number** of all possible ways to colour k vertices by m colours. In section six we introduced a result about the range of edges in which we can colour k vertices bym colours. By this result we can determinethe range of edges in which we can colour a huge **number** of vertices bya huge **number** of colours.

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a) The independent **number** of conjugate graph of 𝐺 , α 𝐺 b) The chromatic **number** of conjugate graph of 𝐺 , 𝐺 c) The **clique** **number** of conjugate graph of 𝐺 , 𝐺 d) The dominating **number** of conjugate graph of 𝐺 , γ 𝐺

The intersection graph of subgroups of a group G is a graph whose vertex set is the set of all proper subgroups of G and two distinct vertices are adjacent if and only if their intersection is non-trivial. In this paper, we obtain the **clique** **number** and degree of vertices of intersection graph of subgroups of dihedral group, quaternion group and quasi-dihedral group.

of a graph is its maximal complete subgraph and the **number** of vertices in the largest **clique** of a graph G, denoted by ω(G), is called the **clique** **number** of G. For a graph G = (V, E), a set S ⊆ V is an independent if no two vertices in S are adjacent. The independence **number** α(G) is the maximum size of an independent set in G. The (open) neighbourhood N (a) of a vertex a ∈ V is the set of vertices which are adjacent to a. For each S ⊆ V , N (S) = S

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The chromatic **number** of a graph G, denoted by χ(G), is the minimum **number** of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. An independent set in a graph is a set of vertices no two of which are adjacent. An independent set in a graph is maximum if the graph contains no larger independent set and maximal if the set cannot be extended to a larger independent set; a maximum independent set is necessarily maximal, but not conversely. The cardinality of any maximum independent set in a graph G is called the independent **number** of G and is denoted by α(G). A **clique** in a graph is a set of mutually adjacent vertices. The maximum size of a **clique** in a graph G is called the **clique** **number** of G and denoted by ω(G). Clearly, a set of vertices S is a **clique** of a simple graph G if and only if it is a stable set of its complement G . In particular, ( G ) ( G ) . For any two nonadjacent vertices x and y in

A subset X of the vertices of the graph Γ is called a **clique** if the induced subgraph on X is a complete graph. The maximum size of a **clique** in a graph Γ is called the **clique** **number** of Γ and denoted by ω(Γ). A subset X of the vertices of Γ is called an independent set if the induced subgraph on X has no edges. The maximum size of an independent set in a graph Γ is called the independence **number** of Γ and denoted by α(Γ). Let k > 0 be an integer. A k-vertex coloring of a graph Γ is an assignment of k colors to the vertices of Γ such that no two adjacent vertices have the same color. The vertex chromatic **number** χ(Γ) of a graph Γ, is the minimum k for which Γ has a k-vertex coloring.

The purpose of this article is to introduce the concept of **clique** transversal dominating sets in graphs. This variant is motivated by grouping a network into subnetworks such that any element of a subnetwork is equally important with others within the same subnetwork. Though plenty of applications of the said variant exist in real life, we list some of them.

Given two graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a subgraph isomorphic to H . Let φ( G , H ) be the smallest possible **number** of parts in an H-decomposition of G. It is easy to see that, if H is nonempty, we have φ( G , H ) = e ( G ) − ν H ( G )( e ( H ) − 1 ) , where

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The results of the experiment depend upon the various values of k as shown in Fig. 5. After developing the proposed movie recommendation system using improved k-cliques, the **number** of movies that were to be rated by the new user among the mov- ies recommended by the system was predicted. This paper adopts the most widely used evaluation metric for performance comparison of the proposed recommendation sys- tem. The mean absolute percentage error (MAPE) is a method of prediction accuracy of a forecasting method in statistics that is defined by the formula [31–33]:

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To form the network of Ring of Cliques, n such identical cliques are connected together in a ring like manner. It is sts of even **number** of cliques, i.e. n is an even natural **number**. Also, to simplify the mathematical formulation of the network, it is assumed that two different nodes of the cliques are involved in the ring , for example, two different nodes, say Node a and Node b, participate in the ring formation. This assumption makes the calculation of required probabilities easier and more concrete. The network is formed by joining n such cliques together, and therefore in every **clique**, two ferent nodes are involved in ring connections. Under the given setting, Node a and Node b are two distinct nodes, and therefore, the **number** of nodes inside each **clique** has to be

initiated by high status individuals to control their peers, its success depends on the participation of other **clique** members (Xie et al., 2002). Due to its secretive and anonymous nature (Coyne, Robinson, & Nelson, 2010), the costs associated with engaging in relational aggression are far lower than those in overt aggression (Bjorkqvist et al., 1994), making it feasible for both high status and low-status **clique** members to carry out. For instance, in some acts of relational aggression, such as rumor spreading, the identity of the perpetrator may remain unknown to the victim. Further, individuals may fear the consequences (e.g., becoming the next target) of not engaging in relationally aggressive behaviour initiated by **clique** mates (Adler & Adler, 1995; Juvonen & Galvan, 2009). Participation in relational aggression may demonstrate commitment to and solidarity with the group (Adler & Adler, 1995; Garandeau & Cillessen, 2006). If the

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This paper proposes a novel MCMOEA, for overlapping community detection. In MCMOEA, we introduce the maximal-**clique** graph by using a set of maximal cliques as nodes and the links among maximal cliques as edges. Then based on the maximal-**clique** graph, a **clique**-based repre- sentation scheme is proposed. Since two maximal cliques are allowed to share the same nodes of the original graph, overlap is an intrinsic property of the nodes of the maximal- **clique** graph, which exactly characterizes the overlapping communities. Attributing to this property, the new repre- sentation scheme allows MOEAs to handle the overlapping community detection problem in a way similar to that of the separated community detection, such that the optimiza- tion problem is simplified. As a result, MCMOEA could detect overlapping community structure with higher parti- tion accuracy and lower computational cost when compared with the existing algorithms. Experiments on synthetic and real-world networks show the effectiveness and efficiency of MCMOEA. Comparisons with other five representative algorithms also confirm that MCMOEA is competitive and promising.

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Coauthorship networks consist of links among groups of mutually connected authors that form a **clique**. Classic- al approaches using Social Network Analysis indices do not account for this characteristic. We propose two new cohesion indices based on a **clique** approach, and we redefine the network density using an index of variance of density. We have applied these indices to two coauthorship networks, one comprising researchers that published in Mathematics Education journals and the other comprising researchers from a Computational Modeling Graduate Program. A contextualized and comparative analysis was performed to show the applicability and po- tential of the indices for analyzing social networks data.

Our techniques are based on a combination of random walks on expanders and additive **number** theory. Random walks on expanders have been used to amplify the success probability of RP and BPP algo- rithms without using many additional random bits [1, 29, 13]. This yields a disperser for sources with en- tropy rate greater than 1/2 [13]. By using Chernoff bounds for random walks on expanders [21, 31, 52], we can construct extractors in a similar way. However, random walks provably fail when the entropy rate drops below 1/2, so they were not considered relevant for this case.

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Abstract A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the **number** of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x, denoted as supp(x), which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x, we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k- index assignment problem and we show empirically how the separation algorithms exploiting sparsity improve on existing ones by running them on the largest instances reported in the literature.

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It is important to recall for this context that non-isomorphic graphs may display the same spectral properties, and these are referred to in the literature as cospectral graphs [32]. Fig. 5.1 shows two graphs with cospectral adjacency matrices, and Ta- ble 5.1 shows the **number** of cospectral graphs and their fraction of the total **number** of graphs for graphs of various sizes [32]. For completeness, we should note that in the table, the fraction of cospectral graphs shown, is an overestimate of the po- tential **number** of problem cases that our modified probabilistic relaxation may run into. The problem cases for the approach occur only when two non-isomorphic atoms are cospectral. The numbers shown in the table account for all possible cospectral graphs, many of which may never be encountered as atoms. Recall that an atom is a non-separable component, while many of the cospectral graphs do not satisfy this property. It would be interesting to search for the **number** of cospectral graphs that may be encountered as atoms, thus causing potential problems for our approach.

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In the **clique** matrix a row with all the entries one represents a strongly connected node and it form cliques with all other vertices of the graph. Here in this example the vertex is compatible with all other streams and hence can be allowed to move simultaneously through the intersection with the other streams at the same time. On the other hand, a row with all the entries zero represents a node that does not form **clique** i.e. is not compatible with other streams of the intersection and have to be controlled independently by allotting different cycle time.

When an audio signal is recorded by a microphone in a closed room, it reaches the microphone via not only a direct path, but also infinitely many reverberant paths. The source sound wave is delayed in time and its energy is absorbed by walls when it is delivered by a reverber- ant path. In order to make the problem practically tract- able, the time delay is usually limited to a certain **number** by which the signal energy may almost disap- pear through repeated reflections. The signal recorded by a digital microphone can then be modeled by a dis- crete convolution of a finite impulse response (FIR) fil- ter and the source signal [1-3]. When there are multiple

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Understanding which general techniques can be used to show lower bounds for proof systems is of paramount importance in proof complexity. For propositional proof systems we have a **number** of very effective techniques, most notably the size-width technique of Ben-Sasson and Wigderson [BSW01], deriving size from width bounds, game characteri- sations (e.g. [Pud00, BK14]), the approach via proof-complexity generators (cf. [Kra11]), and feasible interpolation. Feasible interpolation, first introduced by Kraj´ıˇ cek [Kra97], is a particularly successful paradigm that transfers circuit lower bounds to size of proof lower bounds. The technique has been shown to be effective for resolution [Kra97], cutting planes [Pud97] and even strong Frege systems for modal and intuitionistic logics [Hru09]. However, feasible interpolation fails for strong propositional systems as Frege systems under plausible cryptographic and **number**-theoretic assumptions [KP98, BPR00, BDG + 04].

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