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We have a regular graph $G$ of degree $m$ on $n$ vertices and we label each of its vertices with the number $1$ through $n$. What can we say about the maximum difference between the numbers of two adjacent vertices. What is the maximum and minimum, what is the minimum for a specific graph?

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If I understand your question correctly, I believe the quantity you are asking about is known as graph bandwidth. Perhaps the references on that wiki page will be helpful? –  mhum May 19 at 23:18
    
Yup, that's exactly what I want, In fact I thought of this because of this question: artofproblemsolving.com/Forum/… –  user4140 May 20 at 0:59

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