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| {{Graph families defined by their automorphisms}}
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| In [[mathematics]], a '''distance-regular graph''' is a [[Regular graph|regular]] [[Graph (mathematics)|graph]] such that for any two vertices ''v'' and ''w'', the number of vertices at [[distance (graph theory)|distance]] ''j'' from ''v'' and at distance ''k'' from ''w'' depends only upon ''j'', ''k'', and ''i = d(v, w)''.
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| In particular, this holds when ''k = 1'': in a distance-regular graph, for any two vertices ''v'' and ''w'' at distance ''i'' the number of vertices adjacent to ''w'' and at distance ''j'' from ''v'' is the same. It turns out that, conversely, this implies the above definition of distance-regularity.<ref name="Brouwer">
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| [[Andries Brouwer|A.E. Brouwer]], A.M. Cohen, and A. Neumaier (1989), ''Distance Regular Graphs''. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5</ref> Therefore, an equivalent definition is that a '''distance-regular graph''' is a graph for which there exist integers b<sub>i</sub>,c<sub>i</sub>,i=0,...,d such that for any two vertices x,y in G and distance i=d(x,y), there are exactly c<sub>i</sub> neighbors of y in G<sub>i-1</sub>(x) and b<sub>i</sub> neighbors of y in G<sub>i+1</sub>(x), where G<sub>i</sub>(x) is the set of vertices y of G with d(x,y)=i (Brouwer et al., p. 434). The array of integers characterizing a distance-regular graph is known as its intersection array. | |
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| Every [[distance-transitive graph]] is distance regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large [[Graph automorphism|automorphism group]].
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| A distance-regular graph with diameter 2 is [[Strongly regular graph|strongly regular]], and conversely (unless the graph is [[Connectivity (graph theory)|disconnected]]).
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| ==Intersection numbers==
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| It is usual to use the following notation for a distance-regular graph ''G''. The number of vertices is ''n''. The number of neighbors of ''w'' (that is, vertices adjacent to ''w'') whose distance from ''v'' is ''i'', ''i'' + 1, and ''i'' − 1 is denoted by ''a<sub>i</sub>'', ''b<sub>i</sub>'', and ''c<sub>i</sub>'', respectively; these are the '''intersection numbers''' of ''G''. Obviously, ''a''<sub>0</sub> = 0, ''c''<sub>0</sub> = 0, and ''b''<sub>0</sub> equals ''k'', the degree of any vertex. If ''G'' has finite diameter, then ''d'' denotes the diameter and we have ''b<sub>d</sub>'' = 0. Also we have that ''a<sub>i</sub>+b<sub>i</sub>+c<sub>i</sub>= k''
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| The numbers ''a<sub>i</sub>'', ''b<sub>i</sub>'', and ''c<sub>i</sub>'' are often displayed in a three-line array
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| :<math>\left\{\begin{matrix} - & c_1 & \cdots & c_{d-1} & c_d \\ a_0 & a_1 & \cdots & a_{d-1} & a_d \\ b_0 & b_1 & \cdots & b_{d-1} & - \end{matrix}\right\}, </math>
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| called the '''intersection array''' of ''G''. They may also be formed into a [[tridiagonal matrix]]
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| :<math>B:= \begin{pmatrix} a_0 & b_0 & 0 & \cdots & 0 & 0 \\
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| c_1 & a_1 & b_1 & \cdots & 0 & 0 \\
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| 0 & c_2 & a_2 & \cdots & 0 & 0 \\
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| \vdots & \vdots & \vdots & & \vdots & \vdots \\
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| 0 & 0 & 0 & \cdots & a_{d-1} & b_{d-1} \\
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| 0 & 0 & 0 & \cdots & c_d & a_d \end{pmatrix} ,</math>
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| called the '''intersection matrix'''.
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| ==Distance adjacency matrices==
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| Suppose ''G'' is a connected distance-regular graph. For each distance ''i'' = 1, ..., ''d'', we can form a graph ''G<sub>i</sub>'' in which vertices are adjacent if their distance in ''G'' equals ''i''. Let ''A<sub>i</sub>'' be the [[adjacency matrix]] of ''G<sub>i</sub>''. For instance, ''A''<sub>1</sub> is the adjacency matrix ''A'' of ''G''. Also, let ''A''<sub>0</sub> = ''I'', the identity matrix. This gives us ''d'' + 1 matrices ''A''<sub>0</sub>, ''A''<sub>1</sub>, ..., A''<sub>d</sub>'', called the '''distance matrices''' of ''G''. Their sum is the matrix ''J'' in which every entry is 1. There is an important product formula:
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| :<math>A A_i = a_i A_i + b_i A_{i+1} + c_i A_{i-1} .</math>
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| From this formula it follows that each ''A<sub>i</sub>'' is a polynomial function of ''A'', of degree ''i'', and that ''A'' satisfies a polynomial of degree ''d'' + 1. Furthermore, ''A'' has exactly ''d'' + 1 distinct [[eigenvalue]]s, namely the eigenvalues of the intersection matrix ''B'',of which the largest is ''k'', the degree.
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| The distance matrices span a [[vector subspace]] of the vector space of all ''n'' × ''n'' real matrices.
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| It is a remarkable fact that the product ''A<sub>i</sub>'' ''A<sub>j</sub>'' of any two distance matrices is a [[linear combination]] of the distance matrices:
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| :<math>A_i A_j = \sum_{k=0}^d p_{ij}^k A_k . </math>
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| This means that the distance matrices generate an [[association scheme]]. The theory of association schemes is central to the study of distance-regular graphs. For instance, the fact that ''A<sub>i</sub>'' is a polynomial function of ''A'' is a fact about association schemes.
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| ==Examples==
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| * [[Complete graph]]s are distance regular with diameter 1 and degree ''v''−1.
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| * [[Cycle graph|Cycle]]s ''C''<sub>2''d''+1</sub> of odd length are distance regular with ''k'' = 2 and diameter ''d''. The intersection numbers ''a''<sub>''i''</sub> = 0, ''b''<sub>''i''</sub> = 1, and ''c''<sub>''i''</sub> = 1, except for the usual special cases (see above) and ''c''<sub>''d''</sub> = 2.
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| * All [[Moore graph]]s, in particular the [[Petersen graph]] and the [[Hoffman-Singleton graph]], are distance regular.
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| * [[Strongly regular graph]]s are distance regular.
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| * The [[odd graph]]s are distance regular.
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| === Cubic distance-regular graphs ===
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| There are 13 distance-regular [[cubic graphs]]: [[complete graph|K<sub>4</sub>]] (or [[tetrahedron]]), [[complete bipartite graph|K<sub>3,3</sub>]], the [[Petersen graph]], the [[cube graph|cube]], the [[Heawood graph]], the [[Pappus graph]], the [[Coxeter graph]], the [[Tutte–Coxeter graph]], the [[dodecahedron#As a graph|dodecahedron]], the [[Desargues graph]], [[Tutte 12-cage]], the [[Biggs–Smith graph]], and the [[Foster graph]].
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| {{reflist|30em}}
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| ==Further reading==
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| * {{cite book|last=Godsil|first=C. D.|authorlink=Chris Godsil|title=Algebraic combinatorics|series=Chapman and Hall Mathematics Series|publisher=Chapman and Hall|location=New York|year=1993|pages=xvi+362|isbn=0-412-04131-6|mr=1220704|ref=harv}}
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| {{DEFAULTSORT:Distance-Regular Graph}}
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| [[Category:Algebraic graph theory]]
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| [[Category:Graph families]]
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| [[Category:Regular graphs]]
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