What is Regular: Definition and 266 Discussions

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 vertex are equal to each other. A regular graph with vertices of degree


{\displaystyle k}
is called a


{\displaystyle k}
‑regular graph or regular graph of degree


{\displaystyle k}
. Also, from the handshaking lemma, a regular graph contains an even number of vertices with odd degree.
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



{\displaystyle K_{m}}
is strongly regular for any


{\displaystyle m}
A theorem by Nash-Williams says that every


{\displaystyle k}
‑regular graph on 2k + 1 vertices has a Hamiltonian cycle.

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  1. Z

    A Regular vs stable orbits in spherically symmetric potentials

    I am struggling with Hamiltonian formulation of classical mechanics. I think I have grasped the idea of canonical transformations, including the idea of angle-action variables and invariant tori in phase space. Still, few points seem to elude my understanding... Let's talk about a particle...
  2. C

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  3. shivajikobardan

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  4. shivajikobardan

    Comp Sci Regular expression to NFA -- confusing regular expressions-:

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  5. shivajikobardan

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  6. shivajikobardan

    Comp Sci Is the DFA Solution for Regular Expressions Accurate?

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  7. moriah

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  8. Monsterboy

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  9. rbh

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  10. Who_w

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  11. MesonMarcus

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  12. LCSphysicist

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  13. Vanilla Gorilla

    I Is there a way to translate a particle's spin into regular motion?

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  14. A

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  15. Julio1

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  16. F

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  17. BlackPowder

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  18. whoohm

    Maple Anyone using Maple in 2019 on regular basis?

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  19. S

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  20. J

    I Why Does Wormhole Travel Take Longer Than a 'Regular' Path?

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  21. S

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  22. S

    How many words of length n are there in this regular language?

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  23. S

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  24. Sorcerer

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  25. C

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  26. R

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  27. B

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  28. I

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  29. M

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  30. lfdahl

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  31. karush

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  32. Absentee

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  33. alan

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  34. S

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  35. M

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  36. Albert1

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  37. M

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  38. M

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  39. dkotschessaa

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  40. M

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  41. M

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  42. A

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  43. kolleamm

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  44. F

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  45. evinda

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  46. Jianphys17

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  47. S

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  48. N

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  49. domainwhale

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