||Quantum Graphs were introduced in 1953 to model the electron probability density distribution of a free pi-electron in conjugated molecules such as naphthalene. A natural problem for Quantum Graphs is to determine the spectrum of the operator since the physical interpretation of this quantity is energy of an electron in the structure. The inverse problem is more interesting and hence the trace formula will be introduced to discuss it.
It is proven that two non-isometric graphs can be isospectral. The distribution connected with the spectrum is computed for four quantum graphs and it is demonstrated how the Euler characteristic of a graph can be calculated from the spectrum. The limiting procedure when the length of an edge tends to zero is considered and it is shown that a single vertex can replace two vertices connected by a disappearing edge. The same procedure applied to direct spectral problems is considered.