This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The study of topological indices – graph invariants that can be used for describing and predicting physicochemical or pharmacological properties of organic compounds – is currently one of the most active research fields in chemical graph theory. In this paper we study the Schultz index and find a relation with the Wiener index of the armchair polyhex nanotubes _{6}

Topological indices are a convenient method of translating chemical constitution into numerical values that can be used for correlations with physical, chemical or biological properties. This method has been introduced by Harold Wiener as a descriptor for explaining the boiling points of paraffins [

A unified approach to the Wiener topological index and its various recent modifications is presented. Among these modifications particular attention is paid to the Hyper-Wiener, Harary, Szeged, Cluj and Schultz indices as well as their numerous variants and generalizations [

The main chemical applications and mathematical properties of this index were established in a series of studies [

Carbon nanotubes, the one-dimensional carbon allotropes, are intensively studied with respect to their promise to exhibit unique physical properties: mechanical, optical electronic etc. [_{6}_{6}

Throughout this paper _{6}

_{6}

_{k}

Also in the graph

To compute the

_{k}_{10} to _{q,}_{0} (see _{1} which consists of vertical polygon lines 0, 1,. . . . . _{10} is one of the vertices in the first row of the graph _{1}. The left part is the graph _{2} which consists of vertical polygon lines (_{2} in the following way:

(See

We must consider two cases:

Also in the graphs _{2} and for 1 ≤.

_{1} and _{2} we have

Now suppose that _{1} we can see that if _{2} we have

All of this distances give us

For other vertices we can convert those to _{10} by changing transfer vertices and apply a similar argument by choosing suitable _{1} and _{2} and compute _{k}

By a straightforward computation (if irem means the positive integer remainder) we can see:

So, by Lemma 1, when 1 ≤

Also in the graph

This leads us to the following corollary:

Now suppose that

Also if

We summarize the above results in the following proposition

_{6}[

Now we are in the position to prove the main result of this section.

_{6}

_{u}_{∈level 1}

Since there are

Finally by replacing

_{6}

The following code is the MAPLE program [_{6}

> restart;with(networks):

> l:=proc(p,q) (*generating the graph *)

local G,i,j,k,ff,cc;G:=new();

for i from 0 to (2*p–1) do

for j from 1 to q do

addvertex(a[i,j],G);

end do;

end do;

for i from 0 to (2*p–1) do

for j from 1 to (q–1) do

addedge ({a[i,j],a[i,j+1]},G);

end do;

end do;

for i from 0 to (2*p–2)/2 do

for k from 1 to iquo(q,2) do

addedge({a[2*i,2*k–1],a[2*i+1,2*k–1]},G);

end do;

end do;

for i from 0 to (2*p–4)/2 do

for k from 1 to iquo(q,2) do

addedge({a[2*i+1,2*k],a[2*i+2,2*k]},G);

end do;

end do;

for ff from 1 to iquo(q,2) do

addedge({a[2*p–1,2*ff],a[0,2*ff]},G);

end do;

if irem(q,2)=1 then

for cc from 0 to 2*p/2–1 do

addedge({a[2*cc,q],a[2*cc+1,q]},G); end do;

end if ;return(G);

end proc:

> m:=l(3,8):(#Graph G:=TUVC_6[2*3,8]#)

> t :=edges(m):

> ii:=vertices(m):

> T := allpairs(m,p):

> Sch:=proc(u)

local b,o,pp;

b:=0;

for o in ii do

for pp in ii do

b:=b+ T[(pp,o)]*(vdegree(o,m)+vdegree(pp,m));

end do;

end do;

return(b/2);

end proc:

> Sch(u); 27648(#The Schultz index of the graph #)

This work was supported by a grant from the Center of Research of Islamic Azad University, Najafabad Branch, Isfahan, Iran.

A _{6}

An armchair polyhex nanotube [

Distances from x01 to all vertices of _{6}

Schultz index of short tubes,

6 | 2 | 6912 | 5 | 2 | 4000 |

6 | 3 | 18366 | 5 | 3 | 10650 |

6 | 4 | 35424 | 5 | 4 | 20720 |

6 | 5 | 58656 | 9 | 5 | 193266 |

10 | 2 | 32000 | 9 | 6 | 288432 |

10 | 5 | 264160 | 9 | 7 | 404514 |

10 | 6 | 393440 | 9 | 8 | 542880 |

10 | 7 | 550560 | 15 | 8 | 2425440 |

10 | 8 | 736960 | 15 | 7 | 1823310 |

10 | 9 | 954400 | 15 | 6 | 1310160 |

Schultz index of long tubes,

p | q | S(G) | p | q | S(G) |
---|---|---|---|---|---|

4 | 4 | 10816 | 3 | 4 | 4752 |

4 | 5 | 18304 | 3 | 5 | 8262 |

4 | 6 | 28352 | 3 | 6 | 13104 |

4 | 7 | 41344 | 3 | 7 | 19494 |

4 | 8 | 57664 | 3 | 8 | 27648 |

10 | 21 | 6810400 | 11 | 11 | 1954502 |

10 | 22 | 7641600 | 11 | 12 | 2371952 |

10 | 23 | 8536800 | 11 | 13 | 2839524 |

10 | 24 | 9498400 | 11 | 14 | 3359312 |

10 | 25 | 10528800 | 11 | 15 | 3935030 |