Graphs are among the most frequently used structures in Computer Science. In this work, we analyze how we can express some important graph properties such as connectivity, acyclicity and the Eulerian and Hamiltonian properties in a modal logic. First, we show that these graph properties are not definable in a basic modal language. Second, we discuss an extension of the basic modal language with fix-point operators, the modal μ-calculus. Unfortunately, even with all its expressive power, the μ-calculus fails to express these properties. This happens because μ-calculus formulas are invariant under bisimulations. Third, we show that it is possible to express some of the above properties in a basic hybrid logic. Fourth, we propose an extension of CTL* with nominals, that we call hybrid-CTL*, and then show that it can express the Hamiltonian property in a better way than the basic hybrid logic. Finally, we introduce a promising way of expressing properties related to edges and use it to express the Eulerian property.