Let G = (V, E, w) be a graph, where V and E are the sets of vertices and edges of the graph, respectively, and w is the function mapping each of the edges in E to a positive edge weight. If G is a directed graph, (u, v) ∈ E is an ordered pair of vertices, u, v ∈ V such that (u, v) is directed from u to v . If G is undirected , (u, v) ∈ E has no orientation and is identical to (v, u) ∈ E. Unless otherwise specified, all **graphs** discussed are multigraphs; they may contain two or more edges connecting the same two vertices as well as edges that connect a vertex to itself. A path in G is a sequence of adjacent vertices in V such that no vertices may repeat with the exception of the endpoints. A cycle in G is a path such that the beginning vertex is the same as the end vertex [Wes01]. A subset of a path p between vertices u and v in p may be denoted as p(u, v). The magnitude of a path, |p|, is the sum of the weights of all of the edges of the path.

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sets, we may assume w.l.o.g. that our input graph is a **planar** graph of degree at most three. Valiant [41] and Tamassia et al. [38] showed that a k-terminal **planar** graph G with n vertices and degree at most three admits an orthogonal region-preserving embedding into some square grid of size O(n) × O(n). By Lemma 7, we know that the resulting graph exactly preserves all terminal **minimum** **cuts** of G. We remark that since the embedding is region-preserving, the outer face of the input graph is embedded to the outer face of the grid. Therefore, all terminals in the embedded graph lie on the outer face of the grid. Performing appropriate edge subdivisions, we can make all the terminals lie on the boundary of some possibly larger grid. Further, we can add dummy non-terminals and zero edge capacities to transform our graph into a full-grid H. We observe that the latter does not aﬀect any terminal min-cut. The above leads to the following:

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Our proposed **approach** to identifying network **cuts** in WSN is closely related to the **minimum** cut problem [19] of a graph, and spectral clustering [20] has become one of the most popular algorithms for solving the **minimum** cut problem, due to its simple implementation based on computing the eigenvectors and eigenvalues of the input matrix. Recent progresses on spectral clustering include the following. A graph-based semi-supervised learning algorithm [21] is used to provide the solution to the **minimum** cut problem based on given labeled data. A semi-supervised framework [22] is proposed for spectral clustering to improve the efficiency of the power method for computing the spectral clustering solution. Some common approaches to clustering problem based on pairwise distances are described in [23], and the theoretical guaranties for spectral clustering methods are provided. Two methods are proposed in [24] to efficiently combine the eigenvectors of the graph Laplacian matrices for studying the problem of clustering with data, and two algorithms are proposed in [25] to further improve the clustering quality for correlated probabilistic **graphs**. Our goal is to detect the fragility of WSN based on the realistic wireless channel model. Compared with the existing cut detection algorithms, our proposed algorithm can effi- ciently find multiple network **cuts** that have a high probability of occurrence, and separate a large number of nodes or critical nodes from the sink nodes? That is, our method can identify more network **cuts** that significantly affect the performance of WSN. Hence, after considering the quality of the underlying wireless links, the degree and different priorities of sensor nodes according to their locations and tasks, we propose a novel spectral clustering **approach** to identify network **cuts** in WSN. The contributions of this work

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The problem of …nding a Hamiltonian path or cycle, while being named after Hamilton (1858), was already described in an earlier paper by Kirkman (1856), who discussed a **planar** graph that is not Hamiltonian (see Biggs, Lloyd and Wilson (1976) for details on the history of this problem). Since these early publications, more than one thousand papers have been published, providing theoretical insights, algorithms or applications of the problem. Among the theoretical insights there are criteria for Hamiltonicity and non-Hamiltonicity based on certain characteristics of a graph (such as connectedness, toughness or the number of edges), for example, stochastic analyses on the frequency of Hamiltonian **graphs**, theorems on the Hamiltonicity of **graphs** that do not contain speci…c subgraphs, and results on the number of di¤erent Hamiltonian cycles that might exists for a particular graph. An overview of the vast amount of literature on these and related topics can be found in Bermond (1978), Gould (1991) and Gould (2003).

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Here, the length of a face is the number of edges on its boundary; a bounded face is a face with finite length; a plane graph has bounded codegree if there is an upper bound on the length of bounded faces. A graph is uniformly isoperimetric if satisfies an isoperimetric inequality of the form | **S** | ≤ f (|∂ **S** |) for all non-empty finite vertex sets **S**, where f : N → N is a monotone increasing, diverging function and ∂ **S** is the set of vertices not in **S** but with a neighbour in **S**.

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Fuzzy **planar** graph is a very sensitive and important subclass of fuzzy graph. In this paper, many types of edges are mentioned but especially two types for our fuzzy **graphs**, namely effective edges and considerable edges. It‟**s** Also, a comparative study of Kuratowski‟**s** **graphs** and between of fuzzy **planar** graph are madden. A very new concept of good effort strong fuzzy **planar** graph is introduced. Some related results are established. These results always have some applications in subway tunnels, routes, oil/gas pipelines designing‟**s**, etc. It is also shown that an image could be represented by a fuzzy **planar** graph with contraction of such that image can be made with the help of fuzzy **planar** graph.

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some of the edges, that is, by replacing the edges by paths having at most their endvertices in common. A graph G is said to null if it has no edge. A clique of G is a complete subgraph of G and the number of vertices in a largest clique of G denoted by ω(G), is called the clique number of G. The chromatic number of G, denoted by χ(G), is the **minimum** number of colors which can be assigned to the vertices of G in such a way that every two adjacent vertices have different colors. A graph G is said to be weakly perfect if ω(G) = χ(G). Moreover G is called perfect if every induced subgraph of G is weakly perfect. Let G and H be two arbitrary **graphs**. By G ∨ H, we denote the join of G and H. A graph G is said to be **planar** if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. For any undefined notation or terminology in graph theory, we refer the reader to [6, 7].

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The embedding of complete bipartite graphs onto grids with a The embedding of complete bipartite graphs onto grids with a minimum grid cutwidth.. minimum grid cutwidth.[r]

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In 2012, Moshkovitz and Shapira [16] considered saturation in d-uniform d-partite hy- pergraphs. When d = 2, this reduces to saturation in bipartite **graphs**. A graph is weakly- saturated if there is some ordering of the missing edges such that with each edge added into the graph a new copy of the forbidden graph becomes a subgraph. They observed that weakly-saturated subgraphs of bipartite **graphs** are smaller then weakly-ordered-saturated subgraphs of bipartite **graphs**, and the same holds for saturated subgraphs of bipartite **graphs** and ordered-saturated subgraphs of bipartite **graphs**. They provided a construction show- ing that sat(K n,n , K `,m ) ≤ (` + m − 2)n −

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Indeed, the implication (A) → (C) of Theorem 1.1 yields a **planar** Cayley graph G of Γ, with a Γ-covariant embedding σ : |G| → **S** 2 every face of which is bounded by a cycle. Moreover, when applying Lemma 6.3 to prove the impli- cation (A) → (B), we use the last statement of Lemma 6.3 to ensure that σ is orientation-preserving. We then apply the last statement of Lemma 5.6 to obtain a co-compact, properly discontinuous action Γ y Y with Y := **S** 2 \(σ(Ω(G))), where the set Z of points of Y with non-trivial stabiliser contains at most one point from the interior of each face of σ. Restricting Γ y Y to X ′ := Y \Z thus yields a free action, in other words, X ′ regularly covers **S** := (Y \Z)/Γ. Since

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[4]. Noga Alon. Restricted colorings of **graphs**. In Surveys in combinatorics, 1993 (Keele), volume 187 of London Math. Soc. Lecture Note Ser., pages 1{33. Cambridge Univ. Press, Cambridge, 1993. [5]. Noga Alon. Degrees and choice numbers. Random Structures Algorithms, 16(4):364{368, 2000. [6]. Kenneth Appel and Wolfgang Haken. Every **planar** map is four colorable. Bull. Amer. Math. Soc.,

We explain the criterion that determines whether #M -partitions is in FP or #P- complete for a given symmetric 4 × 4 matrix M in the next section. Doing this requires the related concept of list M -partitions, also due to Feder et al. [6]. Here, each vertex of the input graph comes with a list of parts in which it is allowed to be placed. More formally, the input to the problem is a graph G = (V, E) and a function L : V → P (D), where P ( · ) denotes the powerset. An M -partition σ of G respects the function L if σ(v) ∈ L(v) for all vertices v ∈ V . The **counting** list M-partitions problem is defined as follows.

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Abstract. The equitable edge chromatic number is the **minimum** number of colors required to color the edges of graph G, for which G has a proper edge coloring and if the number of edges in any two color classes differ by at most one. In this paper, we obtain the equitable edge chromatic number of **S** n , W n , H n and G n .

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Remark In the past few years, a number of upper bounds on the Estrada index of **graphs** have been established by using the algebraic techniques (see, for example, [, , , ]). In comparison to the algebraic techniques, the bounds based on our graph-theoretical method are related to the degree parameters (mainly the maximum degree), which are somewhat diﬀerent from the previous ones. Moreover, our method would be more eﬀec- tive in some cases. For example, in [] de la Peña et al. showed that, for any graph G,

For bounded degree **planar** **graphs**, the Liouville property is quasi-isometry invariant since transience is. An example in [3] shows that Liouville property and almost sure intersection of simple random walks are not quasi-isometry invariants: there exist two **graphs** G 1 and G 2 that are quasi-isometric such that G 1 has the intersection property

. In Section 4 we derive a bet- ter bound for sparse **graphs**. Specifically, we assume that the largest clique has size k, potentially much smaller than n, and give another DP algorithm that requires O k! 2 k k 2 n operations. Using standard routines, both algorithms can be turned into uniform samplers. In Section 5 we report on em- pirical results and draw conclusions as to which algorithm is the fastest for what type of instances. We end in Section 6 by discussing open questions and directions for future research.

Monoclonal antibodies: the discovery in 1975, Köhler and Millstein discovered how to prepare hybridoma's: a new cell type, resulting from the fusion of B-lymphocytes (immune cells of a mouse) with a myeloma (cancer) cell. The hybridoma combines two characteristics of the parent cells: immortality of the cancer cell and specific antibody production of the B-lymphocyte. Since all hybridoma cells derived of this fusion product make the same cell antibody type, these antibodies are called monoclonal (coming from one clone) antibodies. O p **t** i m i **s** m a n d h u p e a b o u **t** u **s** e f u l n e **s** **s** : The monoclonal antibody technology was quickly adopted by scientists in both industry and academia and led to a hype in industry and academia about the almost unlimited usefulness of monoclonal antibodies. Their usefulness in research and diagnostics has by now been proven: it is difficult to find a research laboratory where monoclonal antibodies are not generated for research purposes, in particular for selection of compounds. Industrially, they are used to recognize proteins from

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The Colin deVerdiere-type parameter can be use- ful in computing **minimum** rank or maximum nullity (over the real numbers). A symmetric real matrix M is said to satisfy the Strong Arnold Hypothesis provided there does not exist a nonzero symmetric matrix X satis- fying:

Chapter 3 is focused on graph embeddings, specically embedding **graphs** into (, δ) - super-regular **graphs** (dened in Chapter 2). The main result of this type, about **graphs** of bounded degree, is formally known as the Blow Up Lemma. The Lemma is stated in the rst section, but not proven. The concept of (f, δ) -embeddability is also introduced in this section. The second section of this chapter proves that the complete **graphs** **t** into this concept, while the following section proves that the p -arrangeable **graphs** are (f, δ) -embeddable with f linear in |G| . The nal section is a heuristic discussion on the Burr-Erd®**s** conjecture.

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The example proofs and the results of this thesis share their need for two proofs to determine the existence of a fully polynomial randomized approximation scheme, or FPRAS (more formally defined in (2.3.2)). Namely, proving a reduction from approximate **counting** to almost uniform sampling in polynomial time, and a proof of a polynomial run time bound on the mixing time which defines approximately how long the Markov chain must run to reach the desired distribution (defined in (2.2.9)). When combined, these results show that given a fully polynomial almost uniform sampler, or FPAUS (defined in (2.4)), an FPRAS also exist.

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