Topological Study of (3,6)– and (4,6)–Fullerenes

Abstract

A (3,6)–fullerene is a cubic plane graph whose faces (including the outer face) have sizes 3 or 6. (4,6)–Fullerene graphs are defined analogously by interchanging triangles with quadrangles. (3,6)–Fullerenes have exactly four triangles and (4,6)–fullerenes have exactly 6 quadrangles. The (4,6)–fullerenes are also called boron fullerenes. In this chapter some infinite families of (3,6)–and (4,6)–fullerenes are presented. The modeling of these fullerenes by considering some topological indices is the main part of this chapter. Finally, some open questions are presented.

Appendix 1 Some GAP Programs

Here, two GAP programs are presented which is useful for calculations presented in this chapter. The first program is for computing Wiener index and the second is for eccentric connectivity index. Notice that these GAP programs have to combine with calculations by TopoCluj described in the Sect. 15.1.

A Gap Program for Computing Wiener Index of Fullerenes

f:=function(M)

local l,i,j,id,k,t,max,a,s,w,d,g;

l:=Length(M);id:=0;t:=[];s:=[];w:=0;d:=[];g:=0;

    for k in [1..l]do

     id:=1;

       for i in [1..l-1]do

          for j in [i+1..l] do

           if M[k][j]>M[k][id] then

            id:=j;

           fi;

          od;

       od;

       Add(t,M[k][id]);

    od;

max:=t[1];

    for a in [2..Length(t)] do

     if t[a] > max then

      max:=t[a];

     fi;

    od;

    for a in [1..max] do

       for i in [2..l] do

          for j in [1..i-1] do

           if M[i][j]=a then

           g:=g+1;

           fi;

          od;

       od;

    Add(d,g);

    g:=0;

    od;

for i in [2..l] do

    for j in [1..i-1] do

     w:=w+M[i][j];

    od;

od;

Print("Distans=",d,"\n");

Print("Wiener index=",w,"\n");

Print("****************","\n");

end;

A Gap Program for Computing Eccentric Connectivity Index of Fullerenes

f:=function(M)

local l,i,j,k,t,id,ii,jj,s,a,iii,w,ww;

t:=[];l:=Length(M);id:=0;s:=0;a:=[];w:=0;ww:=0;

for k in [1..l]do

 id:=1;

 for i in [1..l-1]do

  for j in [i+1..l] do

   if M[k][j]>M[k][id] then

   id:=j;

   fi;

  od;

 od;

Add(t,M[k][id]);

od; ####ecentricity vertices of G

for ii in [1..l]do

 for jj in [1..l]do

   if M[ii][jj]=1 then

   s:=s+1;

   fi;

 od;

 Add(a,s);

 s:=0;

od;####degree vertices of G

for iii in [1..Length(t)] do

 w:=t[iii]*a[iii];

 ww:=ww+w;

 w:=0;

od;######ecentricity connectivity index of G

Print("ecentricity=",t,"\n");

 Print("degree=",a,"\n");

    Print("ecentricity connectivity index=",ww,"\n");

     Print("**********************","\n");

end;