Regular graph

Graph where each vertex has the same number of neighbors
Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive 		→ edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other.[1] A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k. Also, from the handshaking lemma, a regular graph contains an even number of vertices with odd degree.

Special cases

Regular graphs of degree at most 2 are easy to classify: a 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of a disjoint union of cycles and infinite chains.

A 3-regular graph is known as a cubic graph.

A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common. The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices.

The complete graph Km is strongly regular for any m.

  • 0-regular graph
    0-regular graph
  • 1-regular graph
    1-regular graph
  • 2-regular graph
    2-regular graph
  • 3-regular graph
    3-regular graph

Existence

The necessary and sufficient conditions for a k {\displaystyle k} regular graph of order n {\displaystyle n} to exist are that n ≥ k + 1 {\displaystyle n\geq k+1} and that n k {\displaystyle nk} is even.

Proof: A complete graph has every pair of distinct vertices connected to each other by a unique edge. So edges are maximum in complete graph and number of edges are ( n 2 ) = n ( n − 1 ) 2 {\displaystyle {\binom {n}{2}}={\dfrac {n(n-1)}{2}}} and degree here is n − 1 {\displaystyle n-1} . So k = n − 1 , n = k + 1 {\displaystyle k=n-1,n=k+1} . This is the minimum n {\displaystyle n} for a particular k {\displaystyle k} . Also note that if any regular graph has order n {\displaystyle n} then number of edges are n k 2 {\displaystyle {\dfrac {nk}{2}}} so n k {\displaystyle nk} has to be even. In such case it is easy to construct regular graphs by considering appropriate parameters for circulant graphs.

Properties

A theorem by Nash-Williams says that every k‑regular graph on 2k + 1 vertices has a Hamiltonian cycle.

Let A be the adjacency matrix of a graph. Then the graph is regular if and only if j = ( 1 , … , 1 ) {\displaystyle {\textbf {j}}=(1,\dots ,1)} is an eigenvector of A.[2] Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to j {\displaystyle {\textbf {j}}} , so for such eigenvectors v = ( v 1 , … , v n ) {\displaystyle v=(v_{1},\dots ,v_{n})} , we have ∑ i = 1 n v i = 0 {\displaystyle \sum _{i=1}^{n}v_{i}=0} .

A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one. The "only if" direction is a consequence of the Perron–Frobenius theorem.[2]

There is also a criterion for regular and connected graphs : a graph is connected and regular if and only if the matrix of ones J, with J i j = 1 {\displaystyle J_{ij}=1} , is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).[3]

Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix k = λ 0 > λ 1 ≥ ⋯ ≥ λ n − 1 {\displaystyle k=\lambda _{0}>\lambda _{1}\geq \cdots \geq \lambda _{n-1}} . If G is not bipartite, then

D ≤ log ⁡ ( n − 1 ) log ⁡ ( λ 0 / λ 1 ) + 1. {\displaystyle D\leq {\frac {\log {(n-1)}}{\log(\lambda _{0}/\lambda _{1})}}+1.} [4]

Generation

Fast algorithms exist to generate, up to isomorphism, all regular graphs with a given degree and number of vertices.[5]

See also

References

  1. ^ Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientific. pp. 29. ISBN 978-981-02-1859-1.
  2. ^ a b Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.
  3. ^ Curtin, Brian (2005), "Algebraic characterizations of graph regularity conditions", Designs, Codes and Cryptography, 34 (2–3): 241–248, doi:10.1007/s10623-004-4857-4, MR 2128333.
  4. ^ [1][citation needed]
  5. ^ Meringer, Markus (1999). "Fast generation of regular graphs and construction of cages" (PDF). Journal of Graph Theory. 30 (2): 137–146. doi:10.1002/(SICI)1097-0118(199902)30:2<137::AID-JGT7>3.0.CO;2-G.

External links