Unicyclic　graphs　possessing　Kekulé　structures　with　minimal　energy　are　considered.　Let　n　and　l　be　the　numbers　of　vertices　of　graph　and　cycle　C　l　contained　in　the　graph,　respectively;　r　and　j　positive　integers.　It　is　mathematically　verified　that　for　n　≥　6　and　l　=　2r　+　1　or　l　=　4j+2,　S　n4　has　the　minimal　energy　in　the　graphs　exclusive　of　Sn3,　where　Sn4　is　a　graph　obtained　by　attaching　one　pendant　edge　to　each　of　any　two　adjacent　vertices　of　C　4　and　then　by　attaching　n/2　-　3　paths　of　length　2　to　one　of　the　two　vertices;　Sn3　is　a　graph　obtained　by　attaching　one　pendant　edge　and　n/2　-　2　paths　of　length　2　to　one　vertex　of　C　3.　In　addition,　we　claim　that　for　6　≤　n　≤　12,　Sn4　has　the　minimal　energy　among　all　the　graphs　considered　while　for　n　≥　14,　S　n3　has　the　minimal　energy.　©　2006　Springer　Science+Business　Media,　Inc.