81,844 research outputs found

    Density theorems for bipartite graphs and related Ramsey-type results

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    In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements

    On Van, R and S entropies of graphenylene

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    Applications in the disciplines of chemistry, pharmaceuticals, communication, physics, and aeronautics all heavily rely on graph theory. To examine the properties of chemical compounds, the molecules are modelled as a graph. A few physical characteristics of the substance, including its boiling point, enthalpy, pi-electron energy, and molecular weight, are related to its geometric shape. Through the resolution of one of the interdisciplinary problems characterizing the structures of benzenoid hydrocarbons and graphenylene, the essay seeks to ascertain the practical applicability of graph theory. The topological index, which displays the correlation of chemical structures using numerous physical, chemical, and biological processes, is an invariant of a molecular graph connected with the chemical structure. Shannon's concept of entropy served as the basis for the graph entropies with topological indices, which are now used to measure the structural information of chemical graphs. Using various graph entropy metrics, the theory of graphs can be used to establish the link between particular chemical structural features. This study uses the appropriate R, S, Van topological indices to introduce some unique degree-based entropy descriptors. Additionally, the graphenylene structure's entropy measurements indicated above were computed

    On Van, R and S entropies of graphenylene

    Get PDF
    Applications in the disciplines of chemistry, pharmaceuticals, communication, physics, and aeronautics all heavily rely on graph theory. To examine the properties of chemical compounds, the molecules are modelled as a graph. A few physical characteristics of the substance, including its boiling point, enthalpy, pi-electron energy, and molecular weight, are related to its geometric shape. Through the resolution of one of the interdisciplinary problems characterizing the structures of benzenoid hydrocarbons and graphenylene, the essay seeks to ascertain the practical applicability of graph theory. The topological index, which displays the correlation of chemical structures using numerous physical, chemical, and biological processes, is an invariant of a molecular graph connected with the chemical structure. Shannon's concept of entropy served as the basis for the graph entropies with topological indices, which are now used to measure the structural information of chemical graphs. Using various graph entropy metrics, the theory of graphs can be used to establish the link between particular chemical structural features. This study uses the appropriate R, S, Van topological indices to introduce some unique degree-based entropy descriptors. Additionally, the graphenylene structure's entropy measurements indicated above were computed

    On Van, R and S entropies of graphenylene

    Get PDF
    Applications in the disciplines of chemistry, pharmaceuticals, communication, physics, and aeronautics all heavily rely on graph theory. To examine the properties of chemical compounds, the molecules are modelled as a graph. A few physical characteristics of the substance, including its boiling point, enthalpy, pi-electron energy, and molecular weight, are related to its geometric shape. Through the resolution of one of the interdisciplinary problems characterizing the structures of benzenoid hydrocarbons and graphenylene, the essay seeks to ascertain the practical applicability of graph theory. The topological index, which displays the correlation of chemical structures using numerous physical, chemical, and biological processes, is an invariant of a molecular graph connected with the chemical structure. Shannon's concept of entropy served as the basis for the graph entropies with topological indices, which are now used to measure the structural information of chemical graphs. Using various graph entropy metrics, the theory of graphs can be used to establish the link between particular chemical structural features. This study uses the appropriate R, S, Van topological indices to introduce some unique degree-based entropy descriptors. Additionally, the graphenylene structure's entropy measurements indicated above were computed

    On the refined counting of graphs on surfaces

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    Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g in MHV rules for scattering amplitudes, as well as in ordinary QED. Their counting is a special case of the counting of bi-partite embedded graphs. We review and extend relevant mathematical literature and present results on the counting of some infinite classes of bi-partite graphs. Permutation groups and representations as well as double cosets and quotients of graphs are useful mathematical tools. The counting results are refined according to data of physical relevance, such as the structure of the vertices, faces and genus of the embedded graph. These counting problems can be expressed in terms of observables in three-dimensional topological field theory with S_d gauge group which gives them a topological membrane interpretation.Comment: 57 pages, 12 figures; v2: Typos corrected; references adde

    Locally finite graphs with ends: A topological approach, I. Basic theory

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    AbstractThis paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested
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