The　recently　introduced　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.　Furtula　et　al.　determined　the　extremal　ABC　values　for　chemical　trees,　and　showed　that　the　star　tree　S-n　has　the　maximal　ABC　index　among　all　trees.　In　this　work,　we　show　that　among　all　n-vertex　graphs　with　vertex　connectivity　k,　the　graph　K-k　boolean　OR　(K-1　boolean　OR　Kn-　k-1)　is　the　unique　graph　with　maximum　ABC　index.　Furthermore,　we　determine　the　maximum　ABC　index　of　a　connected　graph　with　n　vertices　and　matching　number　beta　and　characterize　the　unique　extremal　graph　as　K　(beta)　root　Kn-beta.　(C)　2012　Elsevier　Ltd.　All　rights　reserved.