Mutlu, GokhanUgurlu, Ekin2025-05-112025-05-1120240420-12132391-4661https://doi.org/10.1515/dema-2024-0069https://hdl.handle.net/20.500.12416/9582In this article, we examine two problems: a fractional Sturm-Liouville boundary value problem on a compact star graph and a fractional Sturm-Liouville transmission problem on a compact metric graph, where the orders alpha i {\alpha }_{i} of the fractional derivatives on the ith edge lie in ( 0 , 1 ) (0,1) . Our main objective is to introduce quantum graph Hamiltonians incorporating fractional-order derivatives. To this end, we construct a fractional Sturm-Liouville operator on a compact star graph. We impose boundary conditions that reduce to well-known Neumann-Kirchhoff conditions and separated conditions at the central vertex and pendant vertices, respectively, when alpha i -> 1 {\alpha }_{i}\to 1 . We show that the corresponding operator is self-adjoint. Moreover, we investigate a discontinuous boundary value problem involving a fractional Sturm-Liouville operator on a compact metric graph containing a common edge between the central vertices of two star graphs. We construct a new Hilbert space to show that the operator corresponding to this fractional-order transmission problem is self-adjoint. Furthermore, we explain the relations between the self-adjointness of the corresponding operator in the new Hilbert space and in the classical L 2 {L}<^>{2} space.eninfo:eu-repo/semantics/openAccessFractional Sturm-Liouville OperatorMetric GraphTransmission ConditionFractional-Order DerivativeStar GraphArticle10.1515/dema-2024-00692-s2.0-85211107159