An orientation of a graph G with even number of vertices is Pfaffian if every even cycle C such that G-V(C) has a perfect matching has an odd number of edges directed in either direction of the cycle. The significance of Pfaffian orientations stems from the fact that if a graph G has one, then the number of perfect matchings of G can be computed in polynomial time. There is a classical result of Kasteleyn that every planar graph has a Pfaffian orientation. Little proved an elegant characterization of bipartite graphs that admit a Pfaffian orientation. Robertson, Seymour and Thomas (1999) gave a polynomial-time recognition algorithm to test whether a bipartite graph is Pfaffian by a structural description of bipartite graphs. In this paper, we consider the Pfaffian property of graphs embedding on the orientable surface with genus one (i.e., the torus). Some sufficient conditions for Pfaffian graphs on the torus are obtained. Furthermore, we show that all quadrilateral tilings on the torus are Pfaffian if and only if they are not bipartite graphs.
Motivated by the connection with the genus of the corresponding link and its application on DNA polyhedral links,in this paper,we introduce a parameter s_(max)(G),which is the maximum number of circles of states of the link diagram D(G)corresponding to a plane(positive)graph G.We show that s_(max)(G)does not depend on the embedding of G and if G is a 4-edge-connected plane graph then s_(max)(G)is equal to the number of faces of G,which cover the results of S.Y.Liu and H.P.Zhang as special cases.
The atom-bond connectivity（ABC） index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC（G） =∑ uv∈E（G） √d u＋dv-2 dudv,where du denotes the degree of a vertex u in G.A chemical graph is a graph in which no vertex has degree greater than 4.In this paper,we obtain the sharp upper and lower bounds on ABC index of chemical bicyclic graphs.
