### Abstract

We introduce a sequence d_{n}^{l} (l = 1, 2, . . .) of functions on fraktur G sign_{n} × fraktur G sign_{n}, where fraktur G sign_{n} is the set of all simple, connected, undirected graphs of order n up to isomorphism. We show that when l = 1 or l ≥ n - 1, d_{n}^{l} is a metric on fraktur G sign _{n}. While (fraktur G sign_{n}, d_{n}^{1}) is a totally disconnected metric space that embodies the classical notion of graph isomorphism, (fraktur G sign_{n}, d_{n}^{l}) is a connected metric space whenever l ≥ n - 1. In this paper, we investigate some properties and the relationship between these two spaces. This work was motivated by the problem of virtual path layout in high-speed computer networks, which concerns embedding a specified virtual network into the given physical network in a way that makes optimal use of the physical network resources.

Original language | English (US) |
---|---|

Pages (from-to) | 232-240 |

Number of pages | 9 |

Journal | Aequationes Mathematicae |

Volume | 66 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2003 |

### Fingerprint

### Keywords

- Distance between graphs
- Graph embeddings
- Virtual path layout

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Aequationes Mathematicae*,

*66*(3), 232-240. https://doi.org/10.1007/s00010-003-2687-5

**A metric on the set of connected simple graphs of given order.** / Bhutani, Kiran R.; Khan, Bilal.

Research output: Contribution to journal › Article

*Aequationes Mathematicae*, vol. 66, no. 3, pp. 232-240. https://doi.org/10.1007/s00010-003-2687-5

}

TY - JOUR

T1 - A metric on the set of connected simple graphs of given order

AU - Bhutani, Kiran R.

AU - Khan, Bilal

PY - 2003/12/1

Y1 - 2003/12/1

N2 - We introduce a sequence dnl (l = 1, 2, . . .) of functions on fraktur G signn × fraktur G signn, where fraktur G signn is the set of all simple, connected, undirected graphs of order n up to isomorphism. We show that when l = 1 or l ≥ n - 1, dnl is a metric on fraktur G sign n. While (fraktur G signn, dn1) is a totally disconnected metric space that embodies the classical notion of graph isomorphism, (fraktur G signn, dnl) is a connected metric space whenever l ≥ n - 1. In this paper, we investigate some properties and the relationship between these two spaces. This work was motivated by the problem of virtual path layout in high-speed computer networks, which concerns embedding a specified virtual network into the given physical network in a way that makes optimal use of the physical network resources.

AB - We introduce a sequence dnl (l = 1, 2, . . .) of functions on fraktur G signn × fraktur G signn, where fraktur G signn is the set of all simple, connected, undirected graphs of order n up to isomorphism. We show that when l = 1 or l ≥ n - 1, dnl is a metric on fraktur G sign n. While (fraktur G signn, dn1) is a totally disconnected metric space that embodies the classical notion of graph isomorphism, (fraktur G signn, dnl) is a connected metric space whenever l ≥ n - 1. In this paper, we investigate some properties and the relationship between these two spaces. This work was motivated by the problem of virtual path layout in high-speed computer networks, which concerns embedding a specified virtual network into the given physical network in a way that makes optimal use of the physical network resources.

KW - Distance between graphs

KW - Graph embeddings

KW - Virtual path layout

UR - http://www.scopus.com/inward/record.url?scp=0345759878&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345759878&partnerID=8YFLogxK

U2 - 10.1007/s00010-003-2687-5

DO - 10.1007/s00010-003-2687-5

M3 - Article

VL - 66

SP - 232

EP - 240

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

SN - 0001-9054

IS - 3

ER -