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| [[Image:Graph book sample.gif|right]]
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| In [[graph theory]], a '''book graph''' (often written <math>B_p</math> ) may be any of several kinds of graph.
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| One kind, which may be called a '''quadrilateral book''', consists of ''p'' [[quadrilateral]]s sharing a common edge (known as the "spine" or "base" of the book).<ref>[http://mathworld.wolfram.com/BookGraph.html Eric W. Weisstein, "Book Graph."] From MathWorld–A Wolfram Web Resource.</ref> A book of this type is the [[Cartesian product of graphs|Cartesian product]] of a star and ''K''<sub>2</sub> .
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| A second type, which might be called a '''triangular book''', is the complete tripartite graph ''K''<sub>1,1,''p''</sub>. It is a graph consisting of <math>p</math> [[triangle]]s sharing a common edge.<ref>Lingsheng Shi and Zhipeng Song, Upper bounds on the spectral radius of book-free and/or ''K''<sub>2,l</sub>-free graphs. ''Linear Algebra and its Applications'', vol. 420 (2007), pp. 526–529. {{doi|10.1016/j.laa.2006.08.007}}</ref> A book of this type is a [[split graph]].
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| This graph has also been called a <math>K_e(2,p)</math>.<ref>{{cite journal|last=Erdős|first=Paul|year=1963|title=On the structure of linear graphs|journal=Israel Journal of Mathematics|volume=vol. 1|pages=pp. 156–160|authorlink=Paul Erdős|doi=10.1007/BF02759702}}</ref>
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| Given a graph <math>G</math>, one may write <math>bk(G)</math> for the largest book (of the kind being considered) contained within <math>G</math>.
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| The term "book-graph" has been employed for other uses. Barioli<ref>Francesco Barioli, Completely positive matrices with a book-graph. ''Linear Algebra and its Applications'', vol. 277 (1998), pp. 11–31. {{doi|10.1016/S0024-3795(97)10070-2}}</ref> used it to mean a graph composed of a number of arbitrary subgraphs having two vertices in common. (Barioli did not write <math>B_p</math> for his book-graph.)
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| ==Theorems on books==
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| Denote the [[Ramsey number]] of two (triangular) books by <math>r(B_p,\ B_q).</math>
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| * If <math>1\leq p\leq q</math>, then <math>r(B_p,\ B_q)=2q+3</math> (proved by [[Rousseau]] and [[Sheehan]]).
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| * There exists a constant <math>c=o(1)</math> such that <math>r(B_p,\ B_q)=2q+3</math> whenever <math>q\geq cp</math>.
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| * If <math> p\leq q/6+o(q)</math>, and <math>q</math> is large, the [[Ramsey number]] is given by <math>2q+3</math>.
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| * Let <math>C</math> be a constant, and <math>k = Cn</math>. Then every graph on <math>n</math> vertices and <math>m</math> edges contains a (triangular) <math>B_k</math>.<ref>P. Erdos, [http://projecteuclid.org/euclid.ijm/1255631811 On a theorem of Rademacher-Turán]. ''Illinois Journal of Mathematics'', vol. 6 (1962), pp. 122–127.</ref>
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Book (Graph Theory)}}
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| [[Category:Parametric families of graphs]]
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| [[Category:Planar graphs]]
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