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.
Let Gn(C) be the sandwich semigroup of generalized circulant Boolean matrices with the sandwich matrix C and GC(Jn) the set of all primitive matrices in Gn(C).In this paper,some necessary and sufcient conditions for A in the semigroup Gn(C) to be primitive are given.We also show that GC(Jn) is a subsemigroup of Gn(C).
