Let　G　be　a　simple　graph　with　n(G)　vertices　and　e(G)　edges.　The　elementary　cyclic　number　c(G)　of　G　is　defined　as　c(G)　=　e(G)　-　n(G)+omega(G),　where　omega(G)　is　the　number　of　connected　components　of　G.　The　nullity　of　G,　denoted　by　eta(G),　is　the　multiplicity　of　the　eigenvalue　zero　of　the　adjacency　matrix　of　G.　A　graph　is　leaf-free　if　it　has　no　pendent　vertices.　In　Ma　et　al.　(2016)　proved　that　if　G　is　leaf-free　and　each　component　of　G　contains　at　least　two　vertices,　then　eta(G)　＜=　2c(G),　the　equality　is　attained　if　and　only　if　G　is　the　union　of　disjoint　cycles,　where　each　cycle　has　length　a　multiple　of　4.　In　this　paper,　we　completely　characterize　all　leaf-free　graphs　with　nullity　one　less　than　the　above　upper　bound,　i.e.,　eta(G)　=　2c(G)　-　1.　(C)　2019　Elsevier　B.V.　All　rights　reserved.