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Diameter of General Knödel Graphs

Seyed Reza Musawi, Esameil Nazari Kiashi

TL;DR

It is proved that diam ( W Δ, n ) = 1 + ⌈ n − 2 2 Δ − 2 ⌉ , where Δ ≥ 2 and n ≥ (2Δ – 5)(2Δ – 2) + 4.

Abstract

The Knödel graph $W_{Δ,n}$ is a $Δ$-regular bipartition graph on $n\ge 2^Δ$ vertices and $n$ is an even integer. The vertices of $W_{Δ,n}$ are the pairs $(i,j)$ with $i=1,2$ and $0\le j\le n/2-1$. For every $j$, $0\le j\le n/2-1$, there is an edge between vertex $(1, j)$ and every vertex $(2,(j+2^k-1) \mod (n/2))$, for $k=0,1,\cdots,Δ-1$. In this paper we obtain some formulas for evaluating the distance of vertices of the Knödel graph and by them, we provide the formula $diam(W_{Δ,n})=1+\lceil\frac{n-2}{2^Δ-2}\rceil$ for the diameter of $W_{Δ,n}$, where $n\ge (2Δ-5)(2^Δ-2)+4$.

Diameter of General Knödel Graphs

TL;DR

It is proved that diam ( W Δ, n ) = 1 + ⌈ n − 2 2 Δ − 2 ⌉ , where Δ ≥ 2 and n ≥ (2Δ – 5)(2Δ – 2) + 4.

Abstract

The Knödel graph is a -regular bipartition graph on vertices and is an even integer. The vertices of are the pairs with and . For every , , there is an edge between vertex and every vertex , for . In this paper we obtain some formulas for evaluating the distance of vertices of the Knödel graph and by them, we provide the formula for the diameter of , where .

Paper Structure

This paper contains 4 sections, 18 theorems, 2 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Distances in $W_{\Delta,n}$
  3. Main result
  4. Conclusion

Key Result

Theorem 1.2

gh For any $0<\epsilon<1$ there exists some $N(\epsilon)$ such that for all $n\geqslant N(\epsilon)$, $\Delta<\log n-(1+\epsilon)\log\log n$ and $i>\epsilon n$ we have $2\lfloor\frac{i}{2^{\Delta-1}-1}\rfloor+1\leqslant d(u_0,w)\leqslant 2\lfloor\frac{i}{2^{\Delta-1}-1}\rfloor+3$, where $w\in\{u_i,v

Figures (2)

  • Figure 1: $W_{4,32}$ can be constructed by two copies of $W_{3,16}$
  • Figure 2: $W_{4,26}$ and a shortest $u_0u_5$-path.

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 11 more