Graphs have become very crucial as they can represent a wide variety of systems in different areas. One interesting structure called community in graphs has attracted considerable attention from both academia and industry. Community detection is meaningful, but typically hard in arbitrary networks. A lot of research has been done based on structural information, but we would like to find communities which are not only cohesive but also influential or important. A k-influential community model based on k-core provided by Li, Qin, Yu, and Mao is helpful to discover these cohesive and important communities. They organize the problem as finding top-r most important communities in a given graph. In this thesis, our goal is to detect top-r most important communities using efficient and memory-saving algorithms running on a consumer-grade machine. We analyze two existing algorithms, then propose multiple new efficient algorithms for this problem. To test their performance, we conduct extensive experiments on some real-world graph datasets. Experimental results show that our algorithms are able to compute top-r most important communities within a very reasonable amount of time and space in a consumer-grade machine. / Graduate

A graph Gis said to be an intersection graph of sets from a family of sets if there exists a function ƒ : V(G)→ such that for u,v V(G), (u,v) E(G) ƒ (u) ƒ (v) ≠ . Interval graphs are thus the intersection graphs of closed intervals on the real line and unit interval graphs are the intersection graphs of unit length intervals on the real line. An interval on the real line can be generalized to a “kbox” in Rk.A kbox B =(R1,R2,...,Rk), where each Riis a closed interval on the real line, is defined to be the Cartesian product R1x R2x…x Rk. If each Ri is a unit length interval, we call B a k-cube. Thus, 1-boxes are just closed intervals on the real line whereas 2-boxes are axis-parallel rectangles in the plane. We study the intersection graphs of k-boxes and k-cubes. The parameter boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes. Thus, interval graphs are the graphs with boxicity at most 1 and unit interval graphs are the graphs with cubicity at most 1. These parameters were introduced by F. S.Roberts in 1969. In some sense, the boxicity of a graph is a measure of how different a graph is from an interval graph and in a similar way, the cubicity is a measure of how different the graph is from a unit interval graph. We prove several upper bounds on the boxicity and cubicity of general as well as special classes of graphs in terms of various graph parameters such as the maximum degree, the number of vertices and the bandwidth. The following are some of the main results presented. 1. We show that for any graph G with maximum degree Δ , box(G)≤ Δ 22 . This result implies that bounded degree graphs have bounded boxicity no matter how large the graph might be. 2. It was shown in [18] that the boxicity of a graph on n vertices with maximum degree Δ is O(Δ ln n). But a similar bound does not hold for the average degree davof a graph. [18] gives graphs in which the boxicity is exponentially larger than davln n. We show that even though an O(davln n) upper bound for boxicity does not hold for all graphs, for almost all graphs, boxicity is O(davln n). 3. The ratio of the cubicity to boxicity of any graph shown in [15] when combined with the results on boxicity show that cub(G) is O(Δ ln 2 n) and O(2 ln n) for any graph G on n vertices and with maximum degree . By using a randomized construction, we prove the better upper bound cub(G) ≤ [4(Δ + 1) ln n.] 4. Two results relating the cubicity of a graph to its bandwidth b are presented. First, it is shown that cub(G) ≤ 12(Δ + 1)[ ln(2b)] + 1. Next, we derive the upper bound cub(G) ≤ b + 1. This bound is used to derive new upper bounds on the cubicity of special graph classes like circular arc graphs, cocomparability graphs and ATfree graphs in relation to the maximum degree. 5. The upper bound for cubicity in terms of the bandwidth gives an upper bound of Δ + 1 for the cubicity of interval graphs. This bound is improved to show that for any interval graph G with maximum degree , cub(G) ≤[ log2 Δ] + 4. 6. Scheinerman [54] proved that the boxicity of any outerplanar graph is at most 2. We present an independent proof for the same theorem. 7. Halin graphs are planar graphs formed by adding a cycle connecting the leaves of a tree none of whose vertices have degree 2. We prove that the boxicity of any Halin graph is equal to 2 unless it is a complete graph on 4 vertices, in which case its boxicity is 1.

A t-spanner of a graph G is a subgraph H of G, which the distance between any two vertices in H is at most t times their distance in G. A tree t-spanner of G is a t-spanner which is a tree. In this dissertation, we discuss the t-spanners on trapezoid, permutation, and interval graphs. We first introduce an O(n) algorithm for finding a tree 4-spanner on trapezoid graphs. Then, give an O(n)algorithm for finding a tree 3-spanner on permutation graphs, improving the existed O(n + m) algorithm. Since the class of permutation graphs is a subclass of trapezoid graphs, we can apply the algorithm on permutation graphs to find the approximation of a tree 3-spanner on trapezoid graphs in O(n) time with edge bound 2n. Finally, we show that not all interval graphs have a tree 2-spanner.

Let G be a simple connected graph on 2n vertices with a perfect matching. For 1 ≤ k ≤ n - 1, G is said to be k-extendable if for every matching M of size k in G there is a perfect matching in G containing all the edges of M. A k-extendable graph G is said to be k-critical (k-minimal) if G+uv (G-uv) is not k-extendable for every non-adjacent (adjacent) pair of vertices u and v of G. The problem that arises is that of characterizing k-extendable, k-critical and k-minimal graphs.In Chapter 2, we establish that δ(G) ≥ 1/2(n + k) is a sufficient condition for a bipartite graph G on 2n vertices to be k-extendable. For a graph G on 2n vertices with δ(G) ≥ n + k 1, n - k even and n/2 ≤ k ≤ n - 2, we prove that a necessary and sufficient condition for G to be k-extendable is that its independence number is at most n - k. We also establish that a k-extendable graph G of order 2n has k + 1 ≤ δ(G) n or δ(G) ≥ 2k + 1, 1 ≤ k ≤ n - 1. Further, we establish the existence of a k-extendable graph G on 2n vertices with δ(G) = j for each integer j Є [k + 1, n] u [2k + 1, 2n 1]. For k = n - 1 and n - 2, we completely characterize k-extendable graphs on 2n vertices. We conclude Chapter 2 with a variation of the concept of extendability to odd order graphs.In Chapter 3, we establish a number of properties of k-critical graphs. These results include sufficient conditions for k-extendable graphs to be k-critical. More specifically, we prove that for a k-extendable graph G ≠ K2n on 2n vertices, 2 ≤ k ≤ n - 1, if for every pair of non-adjacent vertices u and v of G there exists a dependent set S ( a subset S of V (G) is dependent if the induced subgraph G[S] has at least one edge) of G-u-v such that o(G-(S u {u,v})) = S, then G is k-critical. Moreover, for k = 2 this sufficient condition is also a necessary condition for non-bipartite graphs. We also establish a ++ / necessary condition, in terms of the minimum degree, for k-critical graphs.We conclude Chapter 3 by completely characterizing k-critical graphs on 2n vertices for k = 1, n - 1 and n - 2.Chapter 4 contains results on k-minimal graphs. These results include necessary and sufficient conditions for k-extendable graphs to be k-minimal. More specifically, we prove that for a k-extendable graph G on 2n vertices, 1 ≤ k ≤ n - 1, the following are equivalent:G is minimalfor every edge e = uv of G there exists a matching M of size k in G-e such that V(M) n {u,v} = ø and for every perfect matching F in G containing M, e Є F.for every edge e = uv of G there exists a vertex set S of G-u-v such that: M(S) ≥ k; o(G-e-S) = S - 2k + 2; and u and v belong to different odd components of G-e-S, where M(S) denotes a maximum matching in G[S].We also establish a necessary condition, in terms of minimum degree, for k-minimal and k-minimal bipartite graphs. In fact, we prove that a k-minimal graph G ≠ K2n on 2n vertices, 1 ≤ k ≤ n - 1, has minimum degree at most n + k - 1. For a k-minimal bipartite graph G ≠ Kn,n , 1 ≤ k ≤ n - 3, we show that δ(G) < ½(n + k).Chapter 1 provides the notation, terminology, general concepts and the problems concerning extendability graphs and (k,t)-critical graphs.

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In recent years harmonic analysis on combinatorial graphs has attracted considerable attention. The interest is stimulated in part by multiple existing and potential applications of analysis on graphs to information theory, signal analysis, image processing, computer sciences, learning theory, and astronomy. My thesis is devoted to sampling, interpolation, approximation, and multi-resolution on graphs. The results in the existing literature concern mainly with these theories on unweighted graphs. My main objective is to extend existing theories and obtain new results about sampling, interpolation, approximation, and multi-resolution on general combinatorial graphs such as directed, undirected and weighted. / Mathematics

Kirchhoff's laws are well-studied for electrical networks with voltage and current sources, and edges marked by resistors. Kirchhoff's voltage law states that the sum of voltages around any circuit of the network graph is zero, while Kirchhoff's current law states that the sum of the currents along any cutset of the network graph is zero. Given a network, these requirements may be encoded by the circuit matrix and cutset matrix of the network graph. The columns of these matrices are indexed by the edges of the network graph, and their row spaces are orthogonal complements. For (chemical or electrochemical) reaction networks, one must naturally study the opposite problem, beginning with the stoichiometric matrix rather than the network graph. This leads to the following question: given such a matrix, what is a suitable graphic rendering of a network that properly visualizes the underlying chemical reactions? Although we can not expect uniqueness, the goal is to prove existence of such a graph for any matrix. Specifically, we study Kirchhoff graphs, originally introduced by Fehribach. Mathematically, Kirchhoff graphs represent the orthocomplementarity of the row space and null space of integer-valued matrices. After introducing the definition of Kirchhoff graphs, we will survey Kirchhoff graphs in the context of several diverse branches of mathematics. Beginning with combinatorial group theory, we consider Cayley graphs of the additive group of vector spaces, and resolve the existence problem for matrices over finite fields. Moving to linear algebra, we draw a number of conclusions based on a purely matrix-theoretic definition of Kirchhoff graphs, specifically regarding the number of vector edges. Next we observe a geometric approach, reviewing James Clerk Maxwell's theory of reciprocal figures, and presenting a number of results on Kirchhoff duality. We then turn to algebraic combinatorics, where we study equitable partitions, quotients, and graph automorphisms. In addition to classifying the matrices that are the quotient of an equitable partition, we demonstrate that many Kirchhoff graphs arise from equitable edge-partitions of directed graphs. Finally we study matroids, where we review Tutte's algorithm for determining when a binary matroid is graphic, and extend this method to show that every binary matroid is Kirchhoff. The underlying theme throughout each of these investigations is determining new ways to both recognize and construct Kirchhoff graphs.

DE LA TESISSi un graf G admet un etiquetament super edge magic, aleshores G es diu que és un graf super edge màgic. La tesis està principalment enfocada a l'estudi del conjunt de grafs que admeten etiquetaments super edge magic així com també a desenvolupar relacions entre aquest tipus d'etiquetaments i altres etiquetaments molt estudiats com ara els etiquetaments graciosos i armònics, entre d'altres. De fet, els etiquetaments super edge magic serveixen com nexe d'unió entre diferents tipus d'etiquetaments, i per tant moltes relacions entre etiquetaments poden ser obtingudes d'aquesta forma. A la tesis també es proposa una nova manera de pensar en la ja famosa conjectura que afirma que tots els arbres admeten un etiquetament super edge magic. Això és, per a cada arbre T trobam un arbre super edge magic T' que conté a T com a subgraf, i l'ordre de T'no és massa gran quan el comparam amb l'ordre de T . Un problema de naturalesa similar al problema anterior, en el sentit que intentam trobar un graf super edge magic lo més petit possible i que contengui a cert tipus de grafs, i que ha estat completament resolt a la tesis es pot enunciar com segueix.Problema: Quin és un graf conexe G super edge magic d'ordre més petit que conté al graf complet &#61486;Kn com a subgraf?.La solució d'aquest problema és prou interessant ja que relaciona els etiquetaments super edge magic amb un concepte clàssic de la teoria aditiva de nombres com són els conjunts de Sidon dèbils, també coneguts com well spread sets.De fet, aquesta no és la única vegada que el concepte de conjunt de Sidon apareix a la tesis. També quan a la tesis es tracta el tema de la deficiència , els conjunts de Sidon són d'una gran utilitat. La deficiencia super edge magic d'un graf és una manera de mesurar quan d'aprop està un graf de ser super edge magic. Tècnicament parlant, la deficiència super edge magic d'un graf &#61486;G es defineix com el mínim número de vèrtexs aillats amb els que hem d'unirG perque el graf resultant sigui super edge magic. Si d'aquesta manera no aconseguim mai que el graf resultant sigui super edge magic, aleshores deim que la deficiència del graf és infinita. A la tesis, calculam la deficiència super edge magic de moltes families importants de grafs, i a més donam alguns resultats generals, sobre aquest concepte.Per acabar aquest document, simplement diré que al llarg de la tesis molts d'exemples que completen la tesis, i que fan la seva lectura més agradable i entenible han estat introduits. / OF THESISIf a graph G admits a super edge magic labeling, then G is called a super edge magic graph. The thesis is mainly devoted to study the set of graphs which admit super edge magic labelings as well as to stablish and study relations with other well known labelings.For instance, graceful and harmonic labelings, among others, since many relations among labelings can be obtained using super edge magic labelings as the link.In the thesis we also provide a new approach to the already famous conjecture that claims that every tree is super edge magic. We attack this problem by finding for any given tree T a super edge magic tree T' that contains T as a subgraph, and the order of T'is not too large if we compare it with the order of T .A similar problem to this one, in the sense of finding small host super edge magic graphs for certain type of graphs, which is completely solved in the thesis, is the following one.Problem: Find the smallest order of a connected super edge magic graph G that contains the complete graph Kn as a subgraph.The solution of this problem has particular interest since it relates super edge magic labelings with the additive number theoretical concept of weak Sidon set, also known as well spread set. In fact , this is not the only time that this concept appears in the thesis.Also when studying the super edge magic deficiency, additive number theory and in particular well spread sets have proven to be very useful. The super edge magic deficiency of graph is a way of measuring how close is graph to be super edge magic.Properly speaking, the super edge magic deficiency of a graph G is defined to be the minimum number of isolated vertices that we have to union G with, so that the resulting graph is super edge magic. If no matter how many isolated vertices we union G with, the resulting graph is never super edge magic, then the super edge magic deficiency is defined to be infinity. In the thesis, we compute the super edge magic deficiency of may important families of graphs and we also provide some general results, involving this concept.Finally, and in order to bring this document to its end, I will just mention that many examples that improve the clarity of the thesis and makes it easy to read, can be found along the hole work.

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