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| {{unsolved|mathematics|Does a Moore graph with girth 5 and degree 57 exist?}}
| | Hello and welcome. My title is Numbers Wunder. What I love doing is taking part in baseball but I haven't made a dime with it. Since she was 18 she's been working as a meter reader but she's always needed her own company. California is our beginning location.<br><br>my blog post; [http://javly.com/?p=56362 javly.com] |
| In [[graph theory]], a '''Moore graph''' is a [[regular graph]] of [[degree (graph theory)|degree]] ''d'' and [[diameter (graph theory)|diameter]] ''k'' whose number of vertices equals the upper bound
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| :<math>1+d\sum_{i=0}^{k-1}(d-1)^i .</math>
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| An equivalent definition of a Moore graph is that it is a graph of diameter ''k'' with [[girth (graph theory)|girth]] 2''k'' + 1. Moore graphs were named by {{harvtxt|Hoffman|Singleton|1960}} after [[Edward F. Moore]], who posed the question of describing and classifying these graphs.
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| As well as having the maximum possible number of vertices for a given combination of degree and diameter,
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| Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a [[cage (graph theory)|cage]] {{harv|Erdõs|Rényi|Sós|1966}}. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth as well as odd girth, and again these graphs are cages.
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| == Bounding vertices by degree and diameter ==
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| [[Image:Petersen-as-Moore.svg|thumb|The [[Petersen graph]] as a Moore graph. Any [[breadth first search]] tree has ''d''(''d''-1)<sup>''i''</sup> vertices in its ''i''th level.]]
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| Let ''G'' be any graph with maximum degree ''d'' and diameter ''k'', and consider the tree formed by [[breadth first search]] starting from any vertex ''v''. This tree has 1 vertex at level 0 (''v'' itself), and at most ''d'' vertices at level 1 (the neighbors of ''v''). In the next level, there are at most ''d''(''d''-1) vertices: each neighbor of ''v'' uses one of its adjacencies to connect to ''v'' and so can have at most ''d''-1 neighbors at level 2. In general, a similar argument shows that at any level 1 ≤ ''i'' ≤ ''k'', there can be at most ''d''(''d''-1)<sup>''i''</sup> vertices. Thus, the total number of vertices can be at most
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| :<math>1+d\sum_{i=0}^{k-1}(d-1)^i.</math>
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| {{harvtxt|Hoffman|Singleton|1960}} originally defined a Moore graph as a graph for which this bound on the number of vertices is met exactly. Therefore, any Moore graph has the maximum number of vertices possible among all graphs with maximum degree ''d'' and diameter ''k''.
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| Later, {{harvtxt|Singleton|1968}} showed that Moore graphs can equivalently be defined as having diameter ''k'' and girth 2''k''+1; these two requirements combine to force the graph to be ''d''-regular for some ''d'' and to satisfy the vertex-counting formula.
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| == Moore graphs as cages ==
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| Instead of upper bounding the number of vertices in a graph in terms of its maximum degree and its diameter, we can calculate via similar methods a lower bound on the number of vertices in terms of its minimum degree and its [[girth (graph theory)|girth]] {{harv|Erdõs|Rényi|Sós|1966}}. Suppose ''G'' has minimum degree ''d'' and girth 2''k''+1. Choose arbitrarily a starting vertex ''v'', and as before consider the breadth-first search tree rooted at ''v''. This tree must have one vertex at level 0 (''v'' itself), and at least ''d'' vertices at level 1. At level 2 (for ''k'' > 1), there must be at least ''d''(''d''-1) vertices, because each vertex at level 1 has at least ''d''-1 remaining adjacencies to fill, and no two vertices at level 1 can be adjacent to each other or to a shared vertex at level 2 because that would create a cycle shorter than the assumed girth. In general, a similar argument shows that at any level 1 ≤ ''i'' ≤ ''k'', there must be at least ''d''(''d''-1)<sup>''i''</sup> vertices. Thus, the total number of vertices must be at least
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| :<math>1+d\sum_{i=0}^{k-1}(d-1)^i.</math>
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| In a Moore graph, this bound on the number of vertices is met exactly. Each Moore graph has girth exactly 2''k''+1: it does not have enough vertices to have higher girth, and a shorter cycle would cause there to be too few vertices in the first ''k'' levels of some breadth first search tree.
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| Therefore, any Moore graph has the minimum number of vertices possible among all graphs with minimum degree ''d'' and diameter ''k'': it is a [[cage (graph theory)|cage]].
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| For even girth 2''k'', one can similarly form a breadth-first search tree starting from the midpoint of a single edge. The resulting bound on the minimum number of vertices in a graph of this girth with minimum degree ''d'' is
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| :<math>2\sum_{i=0}^{k-1}(d-1)^i=1+(d-1)^{k-1}+d\sum_{i=0}^{k-2}(d-1)^i.</math>
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| (The right hand side of the formula instead counts the number of vertices in a breadth first search tree starting from a single vertex, accounting for the possibility that a vertex in the last level of the tree may be adjacent to ''d'' vertices in the previous level.)
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| Thus, the Moore graphs are sometimes defined as including the graphs that exactly meet this bound. Again, any such graph must be a cage.
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| == Examples ==
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| The '''Hoffman–Singleton theorem''' states that any Moore graph with girth 5 must have degree 2, 3, 7, or 57. The Moore graphs are:
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| * The [[complete graph]]s <math> K_n </math> on n > 2 nodes. (diameter 1, girth 3, degree n-1, order n)
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| * The odd [[Cycle graph|cycles]] <math> C_{2n+1} </math>. (diameter n, girth 2n+1, degree 2, order 2n+1)
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| * The [[Petersen graph]]. (diameter 2, girth 5, degree 3, order 10)
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| * The [[Hoffman–Singleton graph]]. (diameter 2, girth 5, degree 7, order 50)
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| * A graph of diameter 2, girth 5, degree 57 and order 3250, whose existence is not known. It is currently unknown whether such a graph exists or is unique.
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| Unlike all other Moore graphs, [[Graham Higman|Higman]] proved that the unknown Moore graph cannot be [[Vertex-transitive graph|vertex-transitive]].
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| If the generalized definition of Moore graphs that allows even girth graphs is used, the Moore graphs also include the even cycles <math> C_{2n} </math>, the [[complete bipartite graph]]s <math> K_{n,n} </math> with girth four, the [[Heawood graph]] with degree 3 and girth 6, and the [[Tutte–Coxeter graph]] with degree 3 and girth 8. More generally, it is known ({{harvnb|Bannai|Ito|1973}}; {{harvnb|Damerell|1973}}) that, other than the graphs listed above, all Moore graphs must have girth 5, 6, 8, or 12.
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| ==See also==
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| * The [[Degree diameter problem]] for graphs and digraphs
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| * [[Table of graphs|Table of degree diameter graphs]]
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| == References ==
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| * {{citation
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| | last = Bollobás | first = Béla | author-link = Béla Bollobás
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| | contribution = Chap.VIII.3
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| | publisher = [[Springer-Verlag]]
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| | series = [[Graduate Texts in Mathematics]]
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| | title = Modern graph theory
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| | volume = 184
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| | year = 1998}}.
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| * {{citation
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| | last1 = Bannai
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| | first1 = E.
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| | last2 = Ito
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| | first2 = T.
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| | title = On finite Moore graphs
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| | url = http://repository.dl.itc.u-tokyo.ac.jp/dspace/handle/2261/6123
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| | journal = Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics
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| | volume = 20
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| | pages = 191–208
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| | year = 1973
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| | mr = 0323615}}
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| * {{citation
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| | last = Damerell
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| | first = R. M.
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| | title = On Moore graphs
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| | journal = Mathematical Proceedings of the Cambridge Philosophical Society
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| | volume = 74
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| | pages = 227–236
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| | year = 1973
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| | mr = 0318004
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| | doi = 10.1017/S0305004100048015}}
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| *{{citation
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| | last1 = Erdõs | first1 = Paul | author1-link = Paul Erdõs
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| | last2 = Rényi | first2 = Alfréd | author2-link = Alfréd Rényi
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| | last3 = Sós | first3 = Vera T. | author3-link = Vera T. Sós
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| | journal = Studia Sci. Math. Hungar.
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| | pages = 215–235
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| | title = On a problem of graph theory
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| | url = http://www.math-inst.hu/~p_erdos/1966-06.pdf
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| | volume = 1
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| | year = 1966}}.
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| * {{citation
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| | author1-link = Alan Hoffman (mathematician)
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| | last1 = Hoffman
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| | first1 = Alan J.
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| | last2 = Singleton
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| | first2 = Robert R.
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| | title = Moore graphs with diameter 2 and 3
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| | journal = IBM Journal of Research and Development
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| | volume = 5
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| | issue = 4
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| | year = 1960
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| | pages = 497–504
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| | doi = 10.1147/rd.45.0497
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| | mr = 0140437}}
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| * {{citation
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| | title = There is no irregular Moore graph
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| | last = Singleton
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| | first = Robert R.
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| | journal = [[American Mathematical Monthly]]
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| | volume = 75
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| | issue = 1
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| | pages = 42–43
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| | year = 1968
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| | mr = 0225679
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| | doi = 10.2307/2315106}}
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| == External links ==
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| * [[Andries Brouwer|Brouwer]] and Haemers: [http://www.cwi.nl/~aeb/math/ipm.pdf Spectra of graphs]
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| * {{MathWorld|urlname=MooreGraph|title=Moore Graph}}
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| * {{MathWorld|urlname=Hoffman-SingletonTheorem|title=Hoffman-Singleton Theorem}}
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| [[Category:Graph families]]
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| [[Category:Regular graphs]]
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Hello and welcome. My title is Numbers Wunder. What I love doing is taking part in baseball but I haven't made a dime with it. Since she was 18 she's been working as a meter reader but she's always needed her own company. California is our beginning location.
my blog post; javly.com