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On the diameter of a super-order-commuting graph

Janko Bračič, Bojan Kuzma

Abstract

We answer a question about the diameter of an order-super-commuting graph on a symmetric group by studying the number-theoretical concept of $d$-complete sequences of primes in arithmetic progression.

On the diameter of a super-order-commuting graph

Abstract

We answer a question about the diameter of an order-super-commuting graph on a symmetric group by studying the number-theoretical concept of -complete sequences of primes in arithmetic progression.
Paper Structure (4 sections, 8 theorems, 13 equations, 3 tables)

This paper contains 4 sections, 8 theorems, 13 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Generating polynomials
  4. Prime numbers congruent $1$, respectively $3$, modulo $4$

Key Result

Proposition 1.1

Let $n\ge 4$. If neither $n$ nor $n-1$ is a prime number, then $\mathop{\mathrm{diam}}\nolimits\Delta^o(\mathcal{S}_n)^\ast=3$ if and only if there exist nonempty disjoint subsets $\mathscr{T}_1,\mathscr{T}_2$ consisting of primes smaller or equal to $n$, such that, for some positive integers $\alph and $p+M^\beta_{\mathscr{T}_2}>n$, for every $p\in \mathscr{T}_1$.∎

Theorems & Definitions (13)

  • Proposition 1.1
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 3 more