# 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

k

{\displaystyle k}
is called a

k

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

k

{\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

K

m

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

m

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

k

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

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1. ### 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. ### Comp Sci Regular Expression in Theory of Automata and Computation

In first part,since every block of 4 consecutive symbol contain at least 2 a's The answer in notes is given (aa(a+b)(a+b)+a(a+b)a(a+b)+a(a+b)(a+b)a+(a+b)aa(a+b)+(a+b)a(a+b)a+(a+b)(a+b)aa)+ But this wont be true since if we choose aabbbbaa which is possible according to the above regular...
3. ### What is [50-99] in regular expression?

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5. ### Comp Sci Regular expression to NFA -- confusing regular expressions-:

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6. ### MHB Regular expression to NFA-confusing regular expressions-:

R=(01+010)* For it I made the below nfa which i believe seems correct. Plus the tutorials that I am following also make sure it’s correct. Q0 is initial state(forgot to mention in figure)...
7. ### Comp Sci Is the DFA Solution for Regular Expressions Accurate?

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8. ### Regular Grade Gas and its Effects on Car Engines

Is regular grade gas detrimental to a car’s engine?
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10. ### Point charges in a regular hexagon

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11. ### Moment of inertia of a regular triangle

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12. ### Can you solve this puzzle involving regular expressions?

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33. ### "Regular" filtration with reverse osmosis filter?

A nanofiltration filter has a pore size around 0.001 micron. Reverse osmosis filters have a pore size around 0.0001 micron. Additionaly, reverse osmosis occurs when a semi-permeable membrane separates solutions of different concentrations so osmotic pressure has to be applied to move water from...
34. ### Derive approximate expression by regular solution theory

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35. S

### Is Electric Toothbrush better then regular toothbrushes?

Hi Everyone! I'm wondering.. Is Electric Toothbrush better then regular toothbrushes? Can you please share thoughts on this? Thanks in advance.
36. ### Variations of Regular Curves problem

Homework Statement Let γs : I → Rn, s ∈ (−δ, δ), > 0, be a variation with compact support K ⊂ I' of a regular curve γ = γ0. Show that there exists some 0 < δ ≤ ε such that γs is a regular curve for all s ∈ (−δ, δ). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of...
37. ### MHB Calculate $\overline{AB}+\overline{AC}$ of Regular Nonagon ABCDEFGHI

$A\,\, regular \,\,nonagon \,\,ABCDEFGHI,\,\,if \,\,\overline{AE}=1$ $find :\overline{AB}+\overline{AC}=?$
38. ### Show Regular Homotopy thru curve and its arclength parameters

Homework Statement Let γ be a regular closed curve in Rn. Show that there is a regular homotopy Γ through closed curves with Γ(−, 0) = γ and Γ(−, 1) an arclength parametrization of γ Homework EquationsThe Attempt at a Solution Hey guys, I just posted another question about homotopy but often...
39. ### Show Regular Homotopy is an Equivalence Relation

Homework Statement Show that regular homotopy of regular curves γ : I → Rn is an equivalence relation, that is: i) γ ∼ γ (where the symbol ∼ stands for “regularly homotopic”); ii) γ ∼ γ˜ implies ˜γ ∼ γ; iii) γ ∼ γ˜ and ˜γ ∼ γˆ implies γ ∼ γˆ (here you have to use a smoothing function)...
40. ### I Etymology of "regular" and "normal" spaces

This is kind of a silly question. I always get "regular" spaces and "normal" spaces confused in topology. This will be a problem if I am asked on a qualifier to prove something about one of these spaces. Is there any linguistic or historical justification to why a regular space deals with a...
41. ### Lattice Models for Fluids - Regular Solution Model

1. Problem Statement: For the regular solution model, develop the equations for the compositions of the coexisting phases in a binary system and plot the phase boundary as a function of χ/RT.2. This question stems from Sandler's Introduction to Applied Statistical Thermodynamics. The Attempt...
42. M

### I A regular matrix <=> mA isomorphism

Hello all Let ##m_A: \mathbb{K^n} \rightarrow \mathbb{K^n}: X \mapsto AX## and ##A \in M_{m,n}(\mathbb{K})## (I already proved that this function is linear) I want to prove that: A regular matrix ##\iff m_A## is an isomorphism. So, here is my approach. Can someone verify whether this is...
43. ### MHB Algebra help - a race around a regular polygon

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44. ### B Could dark matter be regular matter?

What if dark matter is just regular matter that's not very well lit up? What do you think?
45. ### Regular heptagon, finding value of unknowns

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46. ### MHB Proving Regularity of $L^R$ Using DFA Construction

Hello! (Wave) I want to prove that if $L$ is regular then $L^R=\{ w | w^R \in L \}$ is regular. I have thought the following: We suppose that $L$ is regular. Then there is a dfa that recognizes $L$. Assume that $q_0$ is the starting state and $q_n$ is an accepting state, where \$n \in...
47. ### I Do Carmo's book, chap2 Regular surfaces, definition 1.2 -- question

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48. ### Regular expressions with google web search

Hi all, Cam google web search accepet regular expressions? Especially, I need to search strings with any number between two words, for example: "vehicle runs at XXX km/h". Thank you
49. ### Convert data from weird to regular data

Dear Group, I have a 30_sec_data.txt with weird characters in it and data.txt with nice numeric in it, Any one have any ideal to help me convert data from weird to regular data using Matlab. Thank you, Best regard,
50. ### Insights High Temperature Low Temperature Duality for the Ising Model on an Infinite Regular Tree - Comments

domainwhale submitted a new PF Insights post High Temperature Low Temperature Duality for the Ising Model on an Infinite Regular Tree Continue reading the Original PF Insights Post.