Intrinsic Chirality of 3-Connected Graphs

Recently, topology and graph theory have come together to study graphs embedded in 3-space, (see for example [1 5]). An analysis of the symmetries of spatial graphs is useful in chemistry in determining when a molecular bond graph is chiral (i.e., distinct from its mirror image). We are interested in classifying those graphs whose intrinsic structure is such that no matter how they are embedded in space they cannot have mirror image symmetry. Such a graph is said to be intrinsically chiral, since its chirality comes from the intrinsic structure of the graph rather than the extrinsic structure of a particular embedding of the graph in space. The concept of intrinsic chirality also has a chemical interpretation. Stereoisomers are defined to be distinct molecules which have the same chemical formula and whose molecular bond graphs are abstractly isomorphic. For example, a molecule whose bond graph is in the form of a circle and a molecule whose bond graph is in the form of a knot are stereoisomers if they have the same molecular formula. If a molecule is intrinsically chiral then it and all its stereoisomers are chiral and will remain chiral under any thermal agitation. Several chemists have created hierarchies of molecular chirality, in which intrinsic chirality is the strongest type of chirality [6, 7]. Formally, we say that a graph G is intrinsically chiral if, no matter how G is embedded in S (the three sphere), there is no orientation reversing homeomorphism of (S, G). While chemists generally consider graphs embedded in Euclidean 3-space, we can use more topological machinery if article no. 0066