Editing Directed acyclic graph (section)

=== Combinatorial enumeration ===
The [[graph enumeration]] problem of counting directed acyclic graphs was studied by {{harvtxt|Robinson|1973}}.<ref name="enum">{{citation|first=R. W.|last=Robinson|contribution=Counting labeled acyclic digraphs|pages=239–273|editor-first=F.|editor-last=Harary|editor-link=Frank Harary|title=New Directions in the Theory of Graphs|publisher=Academic Press|year=1973}}. See also {{citation
|last1 = Harary | first1 = Frank | author1-link = Frank Harary | first2 = Edgar M. | last2 = Palmer | year =  1973| title = Graphical Enumeration  | publisher = [[Academic Press]] | isbn = 978-0-12-324245-7 | page=19}}.</ref>
The number of DAGs on {{mvar|n}} labeled vertices, for {{math|1=''n''&nbsp;=&nbsp;0, 1, 2, 3, …}} (without restrictions on the order in which these numbers appear in a topological ordering of the DAG) is
:1, 1, 3, 25, 543, 29281, 3781503, … {{OEIS|id=A003024}}.
These numbers may be computed by the [[recurrence relation]]
:<math>a_n = \sum_{k=1}^n (-1)^{k-1} {n\choose k}2^{k(n-k)} a_{n-k}.</math><ref name="enum" />
[[Eric W. Weisstein]] conjectured,<ref>{{MathWorld | urlname=WeissteinsConjecture | title=Weisstein's Conjecture|mode=cs2}}</ref> and {{harvtxt|McKay|Royle|Wanless|Oggier|2004}} proved, that the same numbers count the [[Logical matrix|(0,1) matrices]] for which all [[eigenvalue]]s are positive [[real number]]s. The proof is [[bijective proof|bijective]]: a matrix {{mvar|A}} is an [[adjacency matrix]] of a DAG if and only if {{math|''A''&nbsp;+&nbsp;''I''}} is a (0,1) matrix with all eigenvalues positive, where {{mvar|I}} denotes the [[identity matrix]]. Because a DAG cannot have [[Loop (graph theory)|self-loops]], its adjacency matrix must have a zero diagonal, so adding {{mvar|I}} preserves the property that all matrix coefficients are 0 or 1.<ref>{{citation|last1=McKay|first1=B. D.|author1-link=Brendan McKay (mathematician)|last2=Royle|first2=G. F.|author2-link=Gordon Royle|last3=Wanless|first3=I. M.|last4=Oggier|first4=F. E.|author4-link= Frédérique Oggier |last5=Sloane|first5=N. J. A.|author5-link= Neil Sloane|last6=Wilf|first6=H.|author6-link=Herbert Wilf|title=Acyclic digraphs and eigenvalues of (0,1)-matrices|journal=[[Journal of Integer Sequences]]|volume=7|year=2004|page=33|arxiv=math/0310423|bibcode=2004JIntS...7...33M|url=http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Sloane/sloane15.html}}, Article 04.3.3.</ref>