Moore graph: Difference between revisions

As well as having the maximum possible number of vertices for a given combination of degree and diameter, 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]].{{sfnp|Erdős|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.

[[Image:Petersen-as-Moore.svg|thumb|The [[Petersen graph]] as a Moore graph. Any [[breadth -first search]] tree has {{math|''d''(''d''- − 1)<{{sup>|''i''</sup>−1}}}} vertices in its {{mvar|i}}-th level for {{math|''i''th level&ge; 1}}.]]

Let ''{{mvar|G''}} be any graph with maximum degree ''{{mvar|d''}} and diameter ''{{mvar|k''}}, and consider the tree formed by [[breadth -first search]] starting from any vertex ''{{mvar|v''}}. This tree has 1 vertex at level 0 (''{{mvar|v''}} itself), and at most ''{{mvar|d''}} vertices at level 1 (the neighbors of ''{{mvar|v''}}). In the next level, there are at most {{math|''d''(''d''- − 1)}} vertices: each neighbor of ''{{mvar|v''}} uses one of its adjacencies to connect to ''{{mvar|v''}} and so can have at most {{math|''d''- − 1}} neighbors at level 2. In general, a similar argument shows that at any level {{math|1 ≤ ''i'' ≤ ''k''}}, there can be at most {{math|''d''(''d''- − 1)<{{sup>|''i-1''</sup>−1}}}} vertices. Thus, the total number of vertices can be at most

{{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 ''{{mvar|d''}} and diameter ''{{mvar|k''}}.

Later, {{harvtxt|Singleton|1968}} showed that Moore graphs can equivalently be defined as having diameter ''{{mvar|k''}} and girth {{math|2''k'' + 1}}; these two requirements combine to force the graph to be ''{{mvar|d''}}-regular for some ''{{mvar|d''}} and to satisfy the vertex-counting formula.

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]].{{sfnp|Erdős|Rényi|Sós|1966}} Suppose ''{{mvar|G''}} has minimum degree ''{{mvar|d''}} and girth {{math|2''k'' + 1}}. Choose arbitrarily a starting vertex ''{{mvar|v''}}, and as before consider the breadth-first search tree rooted at ''{{mvar|v''}}. This tree must have one vertex at level 0 (''{{mvar|v''}} itself), and at least ''{{mvar|d''}} vertices at level 1. At level 2 (for {{math|''k'' > 1}}), there must be at least {{math|''d''(''d''- − 1)}} vertices, because each vertex at level 1 has at least {{math|''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 {{math|1 ≤ ''i'' ≤ ''k''}}, there must be at least {{math|''d''(''d''- − 1)<{{sup>|''i''</sup>}}}} vertices. Thus, the total number of vertices must be at least

:<math>1 + d\sum_{i=1}^{k-1}(d-1)^i.</math>

In a Moore graph, this bound on the number of vertices is met exactly. Each Moore graph has girth exactly {{math|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 ''{{mvar|k''}} levels of some breadth -first search tree.

Therefore, any Moore graph has the minimum number of vertices possible among all graphs with minimum degree ''{{mvar|d''}} and diametergirth {{math|2''k'' + 1}}: it is a [[cage (graph theory)|cage]].

For even girth {{math|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 ''{{mvar|d''}} is

:<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>

(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 ''{{mvar|d''}} vertices in the previous level.)

* The [[complete graph]]s <math>{{mvar|K{{sub|n}}}} K_non </{{math> on |''n'' > 2}} nodes (diameter 1, girth 3, degree {{math|''n-'' − 1}}, order {{mvar|n}})

* The odd [[Cycle graph|cycles]] <{{math> C_|''C''{2n{sub|2''n''+1} </math>}}} (diameter {{mvar|n}}, girth 2n{{math|2''n'' + 1}}, degree 2, order 2n{{math|2''n'' + 1}})

* A hypothetical graph (or more than one) of diameter 2, girth 5, degree 57 and order 3250,; whosethe existence of such is unknown and is one of the most famous [[open problem]]s in graph theory.{{sfnp|Dalfó|2019}}

Although all the known Moore graphs are [[Vertexvertex-transitive graph]]s, the unknown one (if it exists) cannot be vertex-transitive, as its [[automorphism group]] can have [[order (group theory)|order]] at most 375, less than its number of vertices.{{sfnp|Mačaj|Širáň|2010}}

If the generalized definition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) [[Generalizedgeneralized polygon]]s. Some examples are the even cycles <{{math> C_|''C''{2n{sub|2''n''}}}} </math>, the [[complete bipartite graph]]s <{{math> K_|''K''{{sub|''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 that, other than the graphs listed above, all Moore graphs must have girth 5, 6, 8, or 12.<ref>{{harvtxt|Bannai|Ito|1973}}; {{harvtxt|Damerell|1973}}</ref> The even girth case also follows from the Feit-Higman theorem about possible values of {{mvar|n}} for a generalized {{mvar|n}}-gon.

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| journal = Linear Algebra and itsIts Applications

| journal = [[Linear Algebra and itsIts Applications]]