Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Diameter of 2-distance graphs

S. H. Jafari, S. R. Musawi

Abstract

For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, for graphs $G$ with diameter 2, we show that $diam(D_2(G))$ can be any integer $t\geqslant2$. For graphs $G$ with $diam(G)\geqslant3$, we prove that $\frac{1}{2}diam(G)\leqslant diam(D_2(G))$ and this inequality is sharp. Also, for $diam(G)=3$, we prove that $diam(D_2(G))\leqslant5$ and this inequality is sharp.

Diameter of 2-distance graphs

Abstract

For a simple graph , the -distance graph, , is a graph with the vertex set and two vertices are adjacent if and only if their distance is in the graph . In this paper, for graphs with diameter 2, we show that can be any integer . For graphs with , we prove that and this inequality is sharp. Also, for , we prove that and this inequality is sharp.
Paper Structure (6 sections, 4 theorems, 1 equation)

This paper contains 6 sections, 4 theorems, 1 equation.

Table of Contents

  1. introduction
  2. Main results
  3. Graphs with diameter 2
  4. A lower bound for diameter of 2-distance graphs
  5. An upper bound for the diameter of 2-distance graphs
  6. Conclusions

Key Result

Lemma 2.4

Let $G$ be a graph and $D_2(G)$ be connected. For any $u,v\in V(G)$ :

Theorems & Definitions (17)

  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 2.8
  • proof
  • proof
  • ...and 7 more