Some topological indices of the family of nanostructures of polycyclic aromatic hydrocarbons (PAHs)

RESEARCH ARTICLE

J.Natn.Sci.Foundation Sri Lanka 2018 46 (1): 81 - 88 DOI: http://dx.doi.org/10.4038/jnsfsr.v46i1.8267

Some topological indices of the family of nanostructures of polycyclic aromatic hydrocarbons (PAHs)

Najmeh Soleimani

^1*

, Esmaeel Mohseni

²

, Niloofar Maleki

²

and Najmeh Imani

²

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Revised: 30 September 2017; Accepted: 19 October 2017

* Corresponding author (soleimani.najmeh@gmail.com; https://orcid.org/0000-0002-2336-1066)

This article is published under the Creative Commons CC-BY-ND License (http://creativecommons.org/licenses/by-nd/4.0/).

This license permits use, distribution and reproduction, commercial and non-commercial, provided that the original work is properly cited and is not changed anyway.

Abstract: 0ROHFXODUGHVFULSWRUVSOD\DVLJQL¿FDQWUROHLQPDQ\

areas such as chemistry and pharmacology. Among these, topological indices have a prominent place. In this paper, we focus on the structure of polycyclic aromatic hydrocarbons and calculate theta, Pi and Sadhana indices of nanostructures. These LQGLFHVRQWKHJURXQGRITXDVLRUWKRJRQDOFXWµTRF¶HGJHVWULSV

in molecular graph. Also, the exact formulae for connectivity indices of nanotubes and nanotori are presented in this manuscript. These topological indices are useful for surveying structure of nanostructures, which have a relationship with degrees of their vertices.

Keywords: Connectivity indices, cut method, nanostructures, Pi index, Sadhana index, theta index.

INTRODUCTION

Graph theory models have extensively been used as predictors of properties of chemical compounds 7ULQDMVWLü  7RGHVFLQL  &RQVRQQL 

DQG UHIHUHQFHV ZLWKLQ 09 'LXGHD ZDV WKH ¿UVW

scientist who considered graph theoretical problems of nanotechnology, and some works on computing topological indices of nanostructure, nanotubes and nanotori are presented in many papers (Diudea & John, 2001; Diudea & Kirby, 2001; Diudea, 2002; Diudea et al., 2003; 2004).

The present study is a continuation of our previous studies (Nikmehr et al., 2015; Soleimani et al., 2015;

2016) on computing some topological indices of nanostructures.

METHODOLOGY

From the point of graph theory, all organic molecular structures can be drawn as graphs in which atoms and bonds are represented by vertices and edges, respectively.

ܸሺܩሻ ܩLet ܧሺܩሻ ݑ ݒ ݑݒ ݀௨ ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ

Ê

ܥଵǡ ܥଶǡ ǥ ǡ ܥ௞

Ê

ܧሺܩሻ

Ê

ܧሺܩሻ ൌ ܥଵ׫ ܥଶ׫ ǥ ׫ ܥ௞

Ê

ܥ௜ת ܥ௝ൌ ׎ǡ

Ê

݅ ് ݆

Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ

ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻ

Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ

Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ

Ê

݉ሺܩǡ ܿሻ

ÊÊ

߯

Ê

݉ ൒ ͳ

ÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜భǥ ݀_௜೘శభǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ

Ê

݅ଵǥ ݅௠ାଵ

Ê

݀௜

Ê

ݒ௜

Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜_భ൅ ڮ ൅ ݀௜_೘శభǤ

௜భǥ௜೘శభ

ǥ ሺͷሻ

Ê

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜_భ݀௜_మ݀௜_యǤ

௜భ௜మ௜య

ǥ ሺ͸ሻ

Ê

ଶܺሺܩሻ ൌ σ _ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔యǤ

௜భ௜మ௜య ǥ ሺ͹ሻ

Ê

ܩ ൌ ܩሾ݌ǡ ݍሿ

Ê

ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ

Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌

Ê

ܩሾʹǡͶሿ

Ê

݁_ଵ

Ê

݁_ଶǡ

Ê

ܿ_ଵǡ

Ê

ܿ_ଷ

Ê

ȁܥሺ݁ଵሻȁ ൌ ͳʹ

Ê

ȁܥሺ݁_ଶሻȁ ൌ ͳͲ

Ê

ȁܥሺ݁ଷሻȁ ൌ ͺ

Ê

ȁܥሺܿଵሻȁ ൌ ʹǡ

Ê

ȁܥሺܿଶሻȁ ൌ Ͷǡ

Ê

ȁܥሺܿଷሻȁ ൌ ͸

Ê

ȁܥሺܿସሻȁ ൌ ͺǤ

Ê

Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ

Ê

ͳͶ

Ê

ܿ_ସ

Ê

ܩ ൌ ܩሾʹǡͶሿ

Ê

be a simple graph with vertex set _{ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀௨ ݀ሺݑǡ ݒሻ}

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻÊ ܥ_ଵǡ ܥ_ଶǡ ǥ ǡ ܥ_௞Ê ܧሺܩሻÊ ܧሺܩሻ ൌ ܥ_ଵ׫ ܥ_ଶ׫ ǥ ׫ ܥ_௞Ê ܥ_௜ת ܥ_௝ൌ ׎ǡÊ ݅ ് ݆Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇ǤÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻÊ

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻÊ

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻÊ

݉ሺܩǡ ܿሻÊÊ ߯Ê ݉ ൒ ͳÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜భǥ ݀_௜೘శభǡ

௜భǥ௜೘శభ

ǥ ሺͶሻÊ

݅ଵǥ ݅௠ାଵÊ ݀௜Ê ݒ௜Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜భ൅ ڮ ൅ ݀௜೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻÊ

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜భ݀௜మ݀௜య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻÊ

ଶܺሺܩሻ ൌ σ _ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔యǤ

௜భ௜మ௜య ǥ ሺ͹ሻÊ ܩ ൌ ܩሾ݌ǡ ݍሿÊ ȁܸሺܩሻȁ ൌ ʹʹ݌ݍÊ ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌Ê ܩሾʹǡͶሿÊ݁ଵÊ ݁ଶǡÊ ܿଵǡÊ ܿଷÊ ȁܥሺ݁ଵሻȁ ൌ ͳʹÊ ȁܥሺ݁ଶሻȁ ൌ ͳͲÊ

ȁܥሺ݁ଷሻȁ ൌ ͺÊ ȁܥሺܿଵሻȁ ൌ ʹǡÊ ȁܥሺܿଶሻȁ ൌ ͶǡÊ ȁܥሺܿଷሻȁ ൌ ͸Ê ȁܥሺܿସሻȁ ൌ ͺǤÊ Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ ͶÊ ͳͶ Ê ܿସÊ ܩ ൌ ܩሾʹǡͶሿÊ

and edge

ܸሺܩሻ ܩ set ܧሺܩሻ ݑ ݒ ݑݒ ݀_௨ ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ

Ê

ܥଵǡ ܥଶǡ ǥ ǡ ܥ௞

Ê

ܧሺܩሻ

Ê

ܧሺܩሻ ൌ ܥଵ׫ ܥଶ׫ ǥ ׫ ܥ௞

Ê

ܥ௜ת ܥ௝ൌ ׎ǡ

Ê

݅ ് ݆

Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ

ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻ

Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ

Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ

Ê

݉ሺܩǡ ܿሻ

ÊÊ

߯

Ê

݉ ൒ ͳ

ÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜భǥ ݀_௜೘శభǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ

Ê

݅ଵǥ ݅௠ାଵ

Ê

݀௜

Ê

ݒ௜

Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜భ൅ ڮ ൅ ݀௜೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ

Ê

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜భ݀_௜మ݀_௜యǤ

௜భ௜మ௜య

ǥ ሺ͸ሻ

Ê

ଶܺሺܩሻ ൌ σ _ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔యǤ

௜_భ௜_మ௜_య ǥ ሺ͹ሻ

Ê

ܩ ൌ ܩሾ݌ǡ ݍሿ

Ê

ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ

Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌

Ê

ܩሾʹǡͶሿ

Ê

݁ଵ

Ê

݁ଶǡ

Ê

ܿଵǡ

Ê

ܿଷ

Ê

ȁܥሺ݁ଵሻȁ ൌ ͳʹ

Ê

ȁܥሺ݁ଶሻȁ ൌ ͳͲ

Ê

ȁܥሺ݁ଷሻȁ ൌ ͺ

Ê

ȁܥሺܿଵሻȁ ൌ ʹǡ

Ê

ȁܥሺܿଶሻȁ ൌ Ͷǡ

Ê

ȁܥሺܿଷሻȁ ൌ ͸

Ê

ȁܥሺܿସሻȁ ൌ ͺǤ

Ê

Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ

Ê

ͳͶ

Ê

ܿସ

Ê

ܩ ൌ ܩሾʹǡͶሿ

Ê

. The edge connecting the vertices

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

ǥ ݀

௜೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

൅ ڮ ൅ ݀

௜_೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜భ

݀

_௜మ

݀

_௜య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

_ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔య

Ǥ

௜భ௜మ௜య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

_ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

_ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

and

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

ǥ ݀

௜_೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

൅ ڮ ൅ ݀

௜೘శభ

Ǥ

௜_భǥ௜_೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜_భ

݀

௜_మ

݀

௜_య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

_ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔య

Ǥ

௜భ௜మ௜య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

_ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

_ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

_ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

_ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

will be denoted by

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

ǥ ݀

௜_೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ Ê

݅

_ଵ

ǥ ݅

_௠ାଵ

Ê ݀

_௜

Ê ݒ

_௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

൅ ڮ ൅ ݀

௜_೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜_భ

݀

௜_మ

݀

௜_య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

_ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔య

Ǥ

௜భ௜మ௜య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

_ଵ

Ê ݁

_ଶ

ǡ Ê ܿ

_ଵ

ǡ Ê ܿ

_ଷ

Ê ȁܥሺ݁

_ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

_ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

ଷ

ሻȁ ൌ ͺ Ê ^ȁܥሺܿ

ଵ

ሻȁ ൌ ʹǡ Ê ^ȁܥሺܿ

ଶ

ሻȁ ൌ Ͷǡ Ê ^ȁܥሺܿ

ଷ

ሻȁ ൌ ͸ Ê ^ȁܥሺܿ

ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

. The degree of the vertex

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

_ଵ

ǡ ܥ

_ଶ

ǡ ǥ ǡ ܥ

_௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

_ଵ

׫ ܥ

_ଶ

׫ ǥ ׫ ܥ

_௞

Ê ܥ

_௜

ת ܥ

_௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

ǥ ݀

_௜೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

൅ ڮ ൅ ݀

௜_೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜_భ

݀

௜_మ

݀

௜_య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

_ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔య

Ǥ

௜భ௜మ௜య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

_ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

, denoted by

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

ǥ ݀

௜೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

൅ ڮ ൅ ݀

_௜೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜భ

݀

௜మ

݀

௜య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

_ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

_ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

_ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

_ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

_ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

_ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

_ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

LVWKHQXPEHURI¿UVWQHLJKERXUVRI

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

ǥ ݀

௜೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

൅ ڮ ൅ ݀

_௜೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜భ

݀

௜మ

݀

௜య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

_ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

_ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

_ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

_ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

_ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

_ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

_ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

in the underlying graph. The distance between

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

ǥ ݀

௜_೘శభ

ǡ

௜_భǥ௜_೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

൅ ڮ ൅ ݀

௜೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜భ

݀

௜మ

݀

௜య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

_ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

_ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

_ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

_ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

_ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

and

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

ǥ ݀

௜_೘శభ

ǡ

௜_భǥ௜_೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

൅ ڮ ൅ ݀

௜೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜భ

݀

௜మ

݀

௜య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

_ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

_ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

_ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

_ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

_ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

in _{ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀௨} ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻÊ ܥଵǡ ܥଶǡ ǥ ǡ ܥ௞Ê ܧሺܩሻÊ ܧሺܩሻ ൌ ܥଵ׫ ܥଶ׫ ǥ ׫ ܥ௞Ê ܥ௜ת ܥ௝ൌ ׎ǡÊ ݅ ് ݆Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇ǤÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻÊ

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻÊ

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻÊ

݉ሺܩǡ ܿሻÊÊ ߯Ê ݉ ൒ ͳÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜_భǥ ݀௜_೘శభǡ

௜_భǥ௜_೘శభ

ǥ ሺͶሻÊ

݅ଵǥ ݅௠ାଵÊ ݀௜Ê ݒ௜Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜_భ൅ ڮ ൅ ݀௜_೘శభǤ

௜_భǥ௜_೘శభ

ǥ ሺͷሻÊ

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜_భ݀௜_మ݀௜_యǤ

௜_భ௜_మ௜_య

ǥ ሺ͸ሻÊ

ଶܺሺܩሻ ൌ σ _ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔యǤ

௜_భ௜_మ௜_య ǥ ሺ͹ሻÊ ܩ ൌ ܩሾ݌ǡ ݍሿÊ ȁܸሺܩሻȁ ൌ ʹʹ݌ݍÊ

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌Ê ܩሾʹǡͶሿÊ݁_ଵÊ ݁_ଶǡÊ ܿ_ଵǡÊ ܿ_ଷÊ ȁܥሺ݁ଵሻȁ ൌ ͳʹÊ ȁܥሺ݁_ଶሻȁ ൌ ͳͲÊ

ȁܥሺ݁ଷሻȁ ൌ ͺÊ ȁܥሺܿଵሻȁ ൌ ʹǡÊ ȁܥሺܿଶሻȁ ൌ ͶǡÊ ȁܥሺܿଷሻȁ ൌ ͸Ê ȁܥሺܿସሻȁ ൌ ͺǤÊ Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ ͶÊ ͳͶ Ê ܿ_ସÊ ܩ ൌ ܩሾʹǡͶሿÊ

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀௨ , ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ

Ê

ܥଵǡ ܥଶǡ ǥ ǡ ܥ௞

Ê

ܧሺܩሻ

Ê

ܧሺܩሻ ൌ ܥଵ׫ ܥଶ׫ ǥ ׫ ܥ௞

Ê

ܥ௜ת ܥ௝ൌ ׎ǡ

Ê

݅ ് ݆

Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ

ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻ

Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ

Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ

Ê

݉ሺܩǡ ܿሻ

ÊÊ

߯

Ê

݉ ൒ ͳ

ÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜_భǥ ݀௜_೘శభǡ

௜_భǥ௜_೘శభ

ǥ ሺͶሻ

Ê

݅ଵǥ ݅௠ାଵ

Ê

݀௜

Ê

ݒ௜

Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜_భ൅ ڮ ൅ ݀௜_೘శభǤ

௜_భǥ௜_೘శభ

ǥ ሺͷሻ

Ê

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜_భ݀௜_మ݀௜_యǤ

௜_భ௜_మ௜_య

ǥ ሺ͸ሻ

Ê

ଶܺሺܩሻ ൌ σ _ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔యǤ

௜_భ௜_మ௜_య ǥ ሺ͹ሻ

Ê

ܩ ൌ ܩሾ݌ǡ ݍሿ

Ê

ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ

Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌

Ê

ܩሾʹǡͶሿ

Ê

݁ଵ

Ê

݁ଶǡ

Ê

ܿଵǡ

Ê

ܿଷ

Ê

ȁܥሺ݁_ଵሻȁ ൌ ͳʹ

Ê

ȁܥሺ݁ଶሻȁ ൌ ͳͲ

Ê

ȁܥሺ݁ଷሻȁ ൌ ͺ

Ê

ȁܥሺܿଵሻȁ ൌ ʹǡ

Ê

ȁܥሺܿଶሻȁ ൌ Ͷǡ

Ê

ȁܥሺܿଷሻȁ ൌ ͸

Ê

ȁܥሺܿସሻȁ ൌ ͺǤ

Ê

Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ

Ê

ͳͶ

Ê

ܿସ

Ê

ܩ ൌ ܩሾʹǡͶሿ

Ê

, is the length of a shortest

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

ǥ ݀

௜೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜భ

൅ ڮ ൅ ݀

_௜೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜భ

݀

௜మ

݀

௜య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

^ଵ

ඥௗ೔భାௗ_೔మାௗ_೔య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

_ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

_ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

_ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

_ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

_ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

_ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

path in

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀௨ ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻÊ ܥଵǡ ܥଶǡ ǥ ǡ ܥ௞Ê ܧሺܩሻÊ ܧሺܩሻ ൌ ܥଵ׫ ܥଶ׫ ǥ ׫ ܥ௞Ê ܥ௜ת ܥ௝ൌ ׎ǡÊ ݅ ് ݆Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇ǤÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻÊ

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻÊ

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻÊ

݉ሺܩǡ ܿሻÊÊ ߯Ê ݉ ൒ ͳÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜భǥ ݀௜೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻÊ

݅ଵǥ ݅௠ାଵÊ ݀௜Ê ݒ௜Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜భ൅ ڮ ൅ ݀_௜೘శభǤ

௜భǥ௜೘శభ

ǥ ሺͷሻÊ

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜భ݀௜మ݀௜య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻÊ

ଶܺሺܩሻ ൌ σ ^ଵ

ඥௗ೔భାௗ_೔మାௗ_೔యǤ

௜భ௜మ௜య ǥ ሺ͹ሻÊ ܩ ൌ ܩሾ݌ǡ ݍሿÊ ȁܸሺܩሻȁ ൌ ʹʹ݌ݍÊ

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌Ê ܩሾʹǡͶሿÊ݁ଵÊ ݁ଶǡÊ ܿଵǡÊ ܿଷÊ ȁܥሺ݁_ଵሻȁ ൌ ͳʹÊ ȁܥሺ݁ଶሻȁ ൌ ͳͲÊ

ȁܥሺ݁ଷሻȁ ൌ ͺÊ ȁܥሺܿ_ଵሻȁ ൌ ʹǡÊ ȁܥሺܿ_ଶሻȁ ൌ ͶǡÊ ȁܥሺܿ_ଷሻȁ ൌ ͸Ê ȁܥሺܿ_ସሻȁ ൌ ͺǤÊ Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ ͶÊ ͳͶ Ê ܿ_ସÊ ܩ ൌ ܩሾʹǡͶሿÊ

. Two edges e = uv and I [\ of _{ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀௨} ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻÊ ܥ_ଵǡ ܥ_ଶǡ ǥ ǡ ܥ_௞Ê ܧሺܩሻÊ ܧሺܩሻ ൌ ܥ_ଵ׫ ܥ_ଶ׫ ǥ ׫ ܥ_௞Ê ܥ_௜ת ܥ_௝ൌ ׎ǡÊ ݅ ് ݆Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇ǤÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻÊ

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻÊ

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻÊ

݉ሺܩǡ ܿሻÊÊ ߯Ê ݉ ൒ ͳÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜భǥ ݀௜೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻÊ

݅ଵǥ ݅௠ାଵÊ ݀௜Ê ݒ௜Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜భ൅ ڮ ൅ ݀_௜೘శభǤ

௜భǥ௜೘శభ

ǥ ሺͷሻÊ

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜భ݀௜మ݀௜య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻÊ

ଶܺሺܩሻ ൌ σ ^ଵ

ඥௗ೔భାௗ_೔మାௗ_೔యǤ

௜భ௜మ௜య ǥ ሺ͹ሻÊ ܩ ൌ ܩሾ݌ǡ ݍሿÊ ȁܸሺܩሻȁ ൌ ʹʹ݌ݍÊ

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌Ê ܩሾʹǡͶሿÊ݁ଵÊ ݁ଶǡÊ ܿଵǡÊ ܿଷÊ ȁܥሺ݁_ଵሻȁ ൌ ͳʹÊ ȁܥሺ݁ଶሻȁ ൌ ͳͲÊ

ȁܥሺ݁ଷሻȁ ൌ ͺÊ ȁܥሺܿ_ଵሻȁ ൌ ʹǡÊ ȁܥሺܿ_ଶሻȁ ൌ ͶǡÊ ȁܥሺܿ_ଷሻȁ ൌ ͸Ê ȁܥሺܿ_ସሻȁ ൌ ͺǤÊ Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ ͶÊ ͳͶ Ê ܿ_ସÊ ܩ ൌ ܩሾʹǡͶሿÊ

are called co-distant, ދe co I¶LIDQGRQO\LIWKH\REH\WKHIROORZLQJUHODWLRQ

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀_௨ ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻÊ ܥଵǡ ܥଶǡ ǥ ǡ ܥ௞Ê ܧሺܩሻÊ ܧሺܩሻ ൌ ܥଵ׫ ܥଶ׫ ǥ ׫ ܥ௞Ê ܥ௜ת ܥ௝ൌ ׎ǡÊ ݅ ് ݆Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇ǤÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻÊ

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻÊ

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻÊ

݉ሺܩǡ ܿሻÊÊ ߯Ê ݉ ൒ ͳÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜_భǥ ݀௜_೘శభǡ

௜భǥ௜೘శభ

ǥ ሺͶሻÊ

݅_ଵǥ ݅_௠ାଵÊ ݀_௜Ê ݒ_௜Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜భ൅ ڮ ൅ ݀_௜೘శభǤ

௜భǥ௜೘శభ

ǥ ሺͷሻÊ

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜భ݀_௜మ݀_௜యǤ

௜భ௜మ௜య

ǥ ሺ͸ሻÊ

ଶܺሺܩሻ ൌ σ _ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔యǤ

௜భ௜మ௜య ǥ ሺ͹ሻÊ ܩ ൌ ܩሾ݌ǡ ݍሿÊ ȁܸሺܩሻȁ ൌ ʹʹ݌ݍÊ ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌Ê ܩሾʹǡͶሿÊ݁ଵÊ ݁_ଶǡÊ ܿ_ଵǡÊ ܿ_ଷÊ ȁܥሺ݁ଵሻȁ ൌ ͳʹÊ ȁܥሺ݁ଶሻȁ ൌ ͳͲÊ

ȁܥሺ݁ଷሻȁ ൌ ͺÊ ȁܥሺܿଵሻȁ ൌ ʹǡÊ ȁܥሺܿଶሻȁ ൌ ͶǡÊ ȁܥሺܿଷሻȁ ൌ ͸Ê ȁܥሺܿସሻȁ ൌ ͺǤÊ Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ ͶÊ ͳͶ Ê ܿସÊ ܩ ൌ ܩሾʹǡͶሿÊ

The above relation coLVUHÀH[LYHDQGV\PPHWULFIRU

any edge e of

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀

௨

 ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ Ê ܥ

ଵ

ǡ ܥ

ଶ

ǡ ǥ ǡ ܥ

௞

Ê ܧሺܩሻ Ê ܧሺܩሻ ൌ ܥ

ଵ

׫ ܥ

ଶ

׫ ǥ ׫ ܥ

௞

Ê ܥ

௜

ת ܥ

௝

ൌ ׎ǡ Ê ݅ ് ݆ Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ

^ଶ

௖

Ǥǥ ሺͳሻ Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ Ê

݉ሺܩǡ ܿሻ ÊÊ ߯ Ê ݉ ൒ ͳ ÊÊ 

௠

߯ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

ǥ ݀

௜_೘శభ

ǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ Ê

݅

ଵ

ǥ ݅

௠ାଵ

Ê ݀

௜

Ê ݒ

௜

Ê 

௠

ܺ ሺܩሻ ൌ ෍ ͳ

ඥ݀

௜_భ

൅ ڮ ൅ ݀

_௜೘శభ

Ǥ

௜_భǥ௜_೘శభ

ǥ ሺͷሻ Ê

ଶ

߯ ሺܩሻ ൌ ෍ ͳ ඥ݀

௜_భ

݀

௜_మ

݀

௜_య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻ Ê

ଶ

ܺ ሺܩሻ ൌ σ

^ଵ

ඥௗ೔భାௗ_೔మାௗ_೔య

Ǥ

௜_భ௜_మ௜_య

ǥ ሺ͹ሻ Ê ܩ ൌ ܩሾ݌ǡ ݍሿ Ê ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌ Ê ܩሾʹǡͶሿ Ê ݁

ଵ

Ê ݁

ଶ

ǡ Ê ܿ

ଵ

ǡ Ê ܿ

ଷ

Ê ȁܥሺ݁

_ଵ

ሻȁ ൌ ͳʹ Ê ȁܥሺ݁

ଶ

ሻȁ ൌ ͳͲ Ê

ȁܥሺ݁

ଷ

ሻȁ ൌ ͺ Ê ȁܥሺܿ

_ଵ

ሻȁ ൌ ʹǡ Ê ȁܥሺܿ

_ଶ

ሻȁ ൌ Ͷǡ Ê ȁܥሺܿ

_ଷ

ሻȁ ൌ ͸ Ê ȁܥሺܿ

_ସ

ሻȁ ൌ ͺǤ Ê Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ Ê ͳͶ Ê ܿ

ସ

Ê ܩ ൌ ܩሾʹǡͶሿ Ê

but in general is not transitive. A graph is called a co-graph if the relation co is also transitive and thus an equivalence relation.

Let

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀_௨ ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻÊ ܥଵǡ ܥଶǡ ǥ ǡ ܥ௞Ê ܧሺܩሻÊ ܧሺܩሻ ൌ ܥଵ׫ ܥଶ׫ ǥ ׫ ܥ௞Ê ܥ௜ת ܥ௝ൌ ׎ǡÊ ݅ ് ݆Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇ǤÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻÊ

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻÊ

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻÊ

݉ሺܩǡ ܿሻÊÊ ߯Ê ݉ ൒ ͳÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜భǥ ݀_௜೘శభǡ

௜భǥ௜೘శభ

ǥ ሺͶሻÊ

݅ଵǥ ݅௠ାଵÊ ݀௜Ê ݒ௜Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜భ൅ ڮ ൅ ݀௜೘శభ

Ǥ

௜భǥ௜೘శభ

ǥ ሺͷሻÊ

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜భ݀௜మ݀௜య

Ǥ

௜భ௜మ௜య

ǥ ሺ͸ሻÊ

ଶܺሺܩሻ ൌ σ _ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔యǤ

௜భ௜మ௜య ǥ ሺ͹ሻÊ ܩ ൌ ܩሾ݌ǡ ݍሿÊ ȁܸሺܩሻȁ ൌ ʹʹ݌ݍÊ ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌Ê ܩሾʹǡͶሿÊ݁ଵÊ ݁_ଶǡÊ ܿ_ଵǡÊ ܿ_ଷÊ ȁܥሺ݁ଵሻȁ ൌ ͳʹÊ ȁܥሺ݁ଶሻȁ ൌ ͳͲÊ

ȁܥሺ݁_ଷሻȁ ൌ ͺÊ ȁܥሺܿଵሻȁ ൌ ʹǡÊ ȁܥሺܿଶሻȁ ൌ ͶǡÊ ȁܥሺܿଷሻȁ ൌ ͸Ê ȁܥሺܿସሻȁ ൌ ͺǤÊ Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ ͶÊ ͳͶ Ê ܿସÊ ܩ ൌ ܩሾʹǡͶሿÊ

be the set of edges

ܸሺܩሻ ܩin ܧሺܩሻ ݑ ݒ ݑݒ ݀௨ ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ

Ê

ܥଵǡ ܥଶǡ ǥ ǡ ܥ௞

Ê

ܧሺܩሻ

Ê

ܧሺܩሻ ൌ ܥଵ׫ ܥଶ׫ ǥ ׫ ܥ௞

Ê

ܥ௜ת ܥ௝ൌ ׎ǡ

Ê

݅ ് ݆

Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ

ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻ

Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ

Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ

Ê

݉ሺܩǡ ܿሻ

ÊÊ

߯

Ê

݉ ൒ ͳ

ÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜భǥ ݀_௜೘శభǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ

Ê

݅ଵǥ ݅௠ାଵ

Ê

݀௜

Ê

ݒ௜

Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜_భ൅ ڮ ൅ ݀௜_೘శభǤ

௜భǥ௜೘శభ

ǥ ሺͷሻ

Ê

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜_భ݀௜_మ݀௜_యǤ

௜భ௜మ௜య

ǥ ሺ͸ሻ

Ê

ଶܺሺܩሻ ൌ σ _ඥௗ ^ଵ

೔భାௗ_೔మାௗ_೔యǤ

௜భ௜మ௜య ǥ ሺ͹ሻ

Ê

ܩ ൌ ܩሾ݌ǡ ݍሿ

Ê

ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ

Ê

ȁܧሺܩሻȁ ൌ ͵͵݌ݍ െ ͷ݌

Ê

ܩሾʹǡͶሿ

Ê

݁_ଵ

Ê

݁_ଶǡ

Ê

ܿ_ଵǡ

Ê

ܿ_ଷ

Ê

ȁܥሺ݁ଵሻȁ ൌ ͳʹ

Ê

ȁܥሺ݁_ଶሻȁ ൌ ͳͲ

Ê

ȁܥሺ݁_ଷሻȁ ൌ ͺ

Ê

ȁܥሺܿଵሻȁ ൌ ʹǡ

Ê

ȁܥሺܿଶሻȁ ൌ Ͷǡ

Ê

ȁܥሺܿଷሻȁ ൌ ͸

Ê

ȁܥሺܿସሻȁ ൌ ͺǤ

Ê

Ͷǡ ͵ǡ ʹǡ Ͷǡ Ͷǡ Ͷ

Ê

ͳͶ

Ê

ܿ_ସ

Ê

ܩ ൌ ܩሾʹǡͶሿ

Ê

that are co-distant to

ܸሺܩሻ ܩ ܧሺܩሻ ݑ ݒ ݑݒ ݀_௨ ݀ሺݑǡ ݒሻ

ݑ̴ݒ ݀ሺݒǡ ݔሻ ൌ ݀ሺݒǡ ݕሻ ൅ ͳ ൌ ݀ሺݑǡ ݔሻ ൅ ͳ ൌ ݀ሺݑǡ ݕሻǤ ܥሺ݁ሻǣ ൌ ሼ݂ א ܧሺܩሻǢ ݂ܿ݋݁ሽ

݁ א ܧሺܩሻ

Ê

ܥ_ଵǡ ܥ_ଶǡ ǥ ǡ ܥ_௞

Ê

ܧሺܩሻ

Ê

ܧሺܩሻ ൌ ܥ_ଵ׫ ܥ_ଶ׫ ǥ ׫ ܥ_௞

Ê

ܥ_௜ת ܥ_௝ൌ ׎ǡ

Ê

݅ ് ݆

Ê

݅ǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ݇Ǥ

ÊÊ 

߆ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ^ଶ

௖

Ǥǥ ሺͳሻ

Ê

ߎሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ܿ ൈ ሺȁܧሺܩሻȁ െ ܿሻ

௖

Ǥǥ ሺʹሻ

Ê

ܵ݀ሺܩሻ ൌ ෍ ݉ሺܩǡ ܿሻ ൈ ሺȁܧሺܩሻȁ െ ܿሻǤ

௖

ǥ ሺ͵ሻ

Ê

݉ሺܩǡ ܿሻ

ÊÊ

߯

Ê

݉ ൒ ͳ

ÊÊ 

௠߯ሺܩሻ ൌ ෍ ͳ

ඥ݀௜భǥ ݀_௜೘శభǡ

௜భǥ௜೘శభ

ǥ ሺͶሻ

Ê

݅ଵǥ ݅௠ାଵ

Ê

݀௜

Ê

ݒ௜

Ê 

௠ܺሺܩሻ ൌ ෍ ͳ

ඥ݀௜భ൅ ڮ ൅ ݀_௜೘శభǤ

௜భǥ௜೘శభ

ǥ ሺͷሻ

Ê

ଶ߯ሺܩሻ ൌ ෍ ͳ ඥ݀௜భ݀_௜మ݀_௜యǤ

௜భ௜మ௜య

ǥ ሺ͸ሻ

Ê

ଶܺሺܩሻ ൌ σ ^ଵ

ඥௗ೔భାௗ_೔మାௗ_೔యǤ

௜_భ௜_మ௜_య ǥ ሺ͹ሻ

Ê

ܩ ൌ ܩሾ݌ǡ ݍሿ

Ê

ȁܸሺܩሻȁ ൌ ʹʹ݌ݍ

Ê