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Fractional Graph Theory: A Rational Approach to the Theory of Graphs
General Theory: Hypergraphs. Fractional Matching. Fractional Coloring. Fractional Edge Coloring. Fractional Arboricity and Matroid Methods. Fractional Isomorphism. Fractional Odds and Ends. Appendix.Expand
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On Random Intersection Graphs: The Subgraph Problem
A new model of random graphs – random intersection graphs – is introduced. In this model, vertices are assigned random subsets of a given set. Two vertices are adjacent provided their assigned setsExpand
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Random Dot Product Graph Models for Social Networks
TLDR
Inspired by the recent interest in combining geometry with random graph models, we explore in this paper two generalizations of the random dot product graph model proposed by Kraetzl, Nickel and Scheinerman, and Tucker [1,2]. Expand
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Representations of Planar Graphs
TLDR
This paper shows that every 3-connected planar graph G can be represented as a collection of circles, one circle representing each vertex and each face, so that, for each edge of G, the four circles representing the two endpoints and the two neighboring faces meet at a point, and furthermore the vertex-circles cross the face-Circles at right angles. Expand
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Fractional isomorphism of graphs
TLDR
We present necessary and sufficient conditions for graphs to be linearly fractionally isomorphic, we prove that quadratic fractional isomorphism is the same as isomorphic and we relate semi-isomorphism to bipartite graphs. Expand
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Degrees of freedom versus dimension for containment orders
Given a family of sets L, where the sets in L admit k ‘degrees of freedom’, we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the followingExpand
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Fractional dimension of partial orders
Given a partially ordered setP=(X, ≤), a collection of linear extensions {L1,L2,...,Lr} is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someLi (andy<x in someLj). For aExpand
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Random intersection graphs when m= w (n): an equivalence theorem relating the evolution of the G ( n, m, p ) and G ( n,P /italic>) models
When the random intersection graph G(n, m, p) proposed by Karonski, Scheinerman, and Singer-Cohen [Combin Probab Comput 8 (1999), 131–159] is compared with the independent-edge G(n, p), theExpand
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Random intersection graphs when m=omega(n): An equivalence theorem relating the evolution of the G(n, m, p) and G(n, p) models
TLDR
When the random intersection graph G(n,m, p) proposed by Karoński, Scheinerman, and Singer-Cohen in [8] is compared with the independent-edge G( n, p), the evolutions are different under some values of m and equivalent under others. Expand
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Modeling graphs using dot product representations
TLDR
Given a simple (weighted) graph, or a collection of graphs on a common vertex set, we seek an assignment of vectors to the vertices such that the dot products of these vectors approximate the weight/frequency of the edges. Expand
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