The spectrum of a quantum graph is defined as the spectrum of the negative second order derivative with Neumann-Kirchhoff boundary conditions, which consists exclusively of its eigenvalues. Speaking of the computation of quantum graph spectra, this always involves the associated eigensolutions of the eigenvalue problem, referred to as eigenfunctions. In this chapter, an algorithm to compute an arbitrarily large part of the lower spectrum of a quantum graph along with a representation of the corresponding eigenfunctions in closed form is proposed. The theoretical foundation is a very remarkable relation of the continuous problem to a nonlinear eigenvalue problem or, in the special case of equilateral graphs, to a linear eigenvalue problem, both only determined on the vertices of the underlying combinatorial graph. Yet this finding does not apply to a particular part of the spectrum, which will be refer to as non-vertex spectrum. Section 5.2 then derives how this relation, nevertheless, can be applied to develop a practical algorithm in the special case of equilateral graphs and for both parts of the spectrum. A generalization to non-equilateral graphs is discussed in Sect. 5.3. New aspects of this chapter are the exploitation of the prominent relation to combinatorial graphs for the whole spectrum using an extended graph, the arising numerical algorithm as well as the application of the latter to solve the nonlinear eigenvalue problem (NEP) in the non-equilateral case.