Due　to　the　capability　to　capture　high-order　information　of　nodes　and　reduce　memory　consumption,　implicit　graph　neural　networks　have　become　an　explored　hotspot　in　recent　years.　However,　these　implicit　graph　neural　networks　are　limited　by　the　static　topology,　which　makes　it　difficult　to　handle　heterophilic　graph-structured　data.　Furthermore,　the　existing　methods　inspired　by　optimization　problem　are　limited　by　the　explicit　structure　of　graph　neural　networks,　which　makes　it　difficult　to　set　an　appropriate　number　of　network　layers　to　solve　optimization　problems.　To　address　these　issues,　we　propose　an　implicit　graph　neural　network　with　flexible　propagation　operators　in　this　paper.　From　the　optimization　objective　function,　we　derive　an　implicit　message　passing　formula　with　flexible　propagation　operators.　Compared　to　the　static　operator,　the　proposed　method　that　joints　the　dynamic　semantic　and　topology　of　data　is　more　applicable　to　heterophilic　graphs.　Moreover,　the　proposed　model　performs　a　fixed-point　iterative　process　for　the　optimization　of　the　objective　function,　which　implicitly　adjusts　the　number　of　network　layers　without　requiring　sufficient　prior　knowledge.　Extensive　experiment　results　demonstrate　the　superiority　of　the　proposed　model.　©　2025　Elsevier　Ltd