Computing the connectivity indices of α- and β-cyclodextrins

Graph theory was successfully applied in developing a relationship between chemical structure and biological activity. The m-connectivity index of a graph G is defined to be
^m
χ(G) = Σ
_v1V2⋯Vm+1
(d
_v
1
d
_V
2
⋯d
_V
m+1
)-1/2 where v. v. ⋯v
_m
. . runs over all paths of length m in G and d
_i
is the degree of vertex v
_i
. Cyclodextrins are a family of compounds widely used in food, pharmaceutical, and chemical industries. In this paper we give explicit formulas for computing the first-, second-, and third-order connectivity indices of the molecular graphs of two classes of cyclodextrins, namely α-cyclodextrin and β-cyclodextrin. These formulas are helpful in the study of the relationship between the structure and activity of these cyclodextrins.