Abstract: Quantum graphs refer to the quantum mechanical problems which are defined on networks formed by some connecting one-dimensional lines. In this paper，some basic concepts as well as the mathematical formulism of the stationary state problem of the quantum graphs are introduced. From the boundary conditions satisfied by the wave functions and their derivatives at vertices，the secular equation determining the energy eigenvalues of a quantum graph is derived. It is proved that the coefficient determinant of the secular equation is real if the number of edges of the quantum graph is even，but is imaginary if the number is odd. This property makes it possible to numerically calculate all the energy levels in a finite energy range. Two typical quantum graphs are taken as examples to analyze the variation of energy eigenvalues with the change of the length of a specific edge.

Key words: quantum graph, eigenstate, energy spectral