thesis title for graph theory

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Title: graph theory and its uses in graph algorithms and beyond.

Abstract: Graphs are fundamental objects that find widespread applications across computer science and beyond. Graph Theory has yielded deep insights about structural properties of various families of graphs, which are leveraged in the design and analysis of algorithms for graph optimization problems and other computational optimization problems. These insights have also proved helpful in understanding the limits of efficient computation by providing constructions of hard problem instances. At the same time, algorithmic tools and techniques provide a fresh perspective on graph theoretic problems, often leading to novel discoveries. In this thesis, we exploit this symbiotic relationship between graph theory and algorithms for graph optimization problems and beyond. This thesis consists of three parts. In the first part, we study a graph routing problem called the Node-Disjoint Paths (NDP) problem. Given a graph and a set of source-destination pairs of its vertices, the goal is to route the maximum number of pairs via node-disjoint paths. We come close to resolving the approximability of NDP by showing that it is $n^{\Omega(1/poly\log\log n)}$-hard to approximate, even on grid graphs, where n is the number of vertices. In the second part of this thesis, we use graph decomposition techniques developed for efficient algorithms to derive a graph theoretic result. We show that for every n-vertex expander graph G, if H is any graph with at most $O(n/\log n)$ vertices and edges, then H is a minor of G. In the last part, we show that the graph theoretic tools and graph algorithmic techniques can shed light on problems seemingly unrelated to graphs. We show that the randomized space complexity of the Longest Increasing Subsequence (LIS) problem in the streaming model is intrinsically tied to the query-complexity of the Non-Crossing Matching problem on graphs in a new model of computation that we define.

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Graph theory: a comprehensive survey about graph theory applications in computer science and social networks.

1. Introduction

1.1. historical background of the graph theory, 1.2. overview of the existing surveys and applications of graph theory in computer science and social networks, 1.3. our contribution, 1.4. paper organization, 2. graph theory practical applications in the computer science field, 2.1. uses of graph theory in algorithms, 2.1.1. girl (graph information retrieval language), 2.1.2. gasp (graph algorithm software package), 2.1.3. gtpl (graph theoretic programming language), 2.2. group special mobile networks and maps coloring using graphs, 2.3. uses of graph algorithms/concepts in network security monitoring, 2.4. services connectivity analysis using graphs, 2.5. web documents clustering using graph theory concepts, 2.6. representing/modelling wireless sensor network as a graph, 2.7. uses of the graph theory in operational research problems, 2.7.1. use of graph coloring, 2.7.2. applications of graph theory in scheduling-related problems, 2.8. applications of graph theory in the internet of things (iot), 2.9. uses of graphs in a blockchin and related technologies, 2.10. uses of graph theory in a computer vision domain/applications, 3. applications of the graph theory in social networks (sn), 3.1. use of graphs for community clustering in a sn, 3.2. use of graphs for information diffusion in social networks, 3.3. users’ influence/trust score representation in a social networks via graphs, 3.4. similarity modeling/representing between social network users via graphs, 3.5. social network analysis via graphs, 3.5.1. closeness centrality, 3.5.2. degree centrality, 3.5.3. betweenness centrality, 3.5.4. eigenvalue centrality, 3.5.5. jordan centrality, 3.6. analyzing the modularity in a social network users’ graph, 3.7. topic of interest modelling using graphs in social networks, 3.8. user identification across social networks by analyzing structural properties of the graphs, 3.9. social-attribute network (san) modelling and analysis via graphs, 3.10. graph theory applications in recommendation systems, 3.11. social interactions modelling between users in social networks via graphs, 3.12. privacy-preserving social network data publishing with researchers/analysts for analysis, 3.13. community-based event detection in temporal networks via graph analysis, 3.14. affiliation network modelling using graphs in social network, 3.15. modelling of the sybil attack in social networks using graphs, 3.16. estimating and inferring the strengths of social relations among different people in sn using graphs, 4. summary and discussion, 5. conclusion and future works, author contributions, acknowledgments, conflicts of interest.

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Majeed, A.; Rauf, I. Graph Theory: A Comprehensive Survey about Graph Theory Applications in Computer Science and Social Networks. Inventions 2020 , 5 , 10. https://doi.org/10.3390/inventions5010010

Majeed A, Rauf I. Graph Theory: A Comprehensive Survey about Graph Theory Applications in Computer Science and Social Networks. Inventions . 2020; 5(1):10. https://doi.org/10.3390/inventions5010010

Majeed, Abdul, and Ibtisam Rauf. 2020. "Graph Theory: A Comprehensive Survey about Graph Theory Applications in Computer Science and Social Networks" Inventions 5, no. 1: 10. https://doi.org/10.3390/inventions5010010

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Topics in Graph Theory

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• Calvin Jongsma 13  

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Graph Theory is an area of modern mathematics with many applications in today’s world, but its roots lie in several recreational puzzles going back to the mid-eighteenth century. This chapter will introduce a few main topics in Graph Theory , drawing upon this history. The first two sections look at ways one can traverse a graph (Eulerian trails and Hamiltonian paths), while the last two sections deal with planar graphs (ones that can be drawn so their edges don’t cross) and graph coloring (graphs whose adjacent vertices have different colors). While connected to a couple of earlier topics, this concluding chapter has a more geometric character, balancing out the algebraic emphasis of the rest of the text.

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See The Truth about Königsberg by Brian Hopkins and Robin J. Wilson in the May 2004 issue, of The College Mathematics Journal for a nice discussion of Euler’s argument. Euler’s paper (and lots more) is in Graph Theory : 1736–1936 (Oxford University Press, 1976) by Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson.

See Joseph Malkevitch’s two 2005 AMS Feature Columns on Euler’s Polyhedral Formula at http://www.ams.org/samplings/feature-column/fcarc-eulers-formula

Robin Wilson’s Four Colors Suffice (Princeton University Press, 2013) gives a fascinating and very readable account of the entire history of the four-color problem.

This is Alexander Soifer’s minimal counterexample.  See The Mathematical Coloring Book (Springer, 2009), p. 182. The Fritsch Graph  provides another counterexample on nine vertices.

For more information on Gardner’s map and its four-coloring, see the Wolfram MathWorld posting on the Four-Color Theorem at http://mathworld.wolfram.com/Four-ColorTheorem.html.

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Jongsma, C. (2019). Topics in Graph Theory. In: Introduction to Discrete Mathematics via Logic and Proof. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-25358-5_8

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