The doctoral dissertation deals with solving three concrete scientific problems from the field of spectral graph theory. First of all, let a nut graph represent a nontrivial simple graph whose adjacency matrix has a one dimensional null space all of whose nonzero members contain no zero elements. The first resolved scientific problem is the circulant nut graph existence problem which is connected to determining all the pairs (n, d), n E N, d E N0 for which there exists a d-regular circulant nut graph of order n. Furthermore, let a balanced tree be a rooted tree all of whose vertices from the same level have an equal number of children. Also, for any d, k E N, d > 2, let the Bethe tree Bd, k represent a balanced tree with k levels such that all of its vertices outside the last level have exactly d - 1 children, and let the dendrimer Dd, k be a balanced tree with k levels such that all of its vertices outside the last level are of degree d. The second scientific problem that the dissertation deals with is the spectral analysis of balanced trees with a special focus on computing the energy of Bethe trees and approximating the energy of dendrimers. Finally, the goal of the third part of the doctoral dissertation is to determine the energy of the newly introduced martini graphs for the purpose of disproving a conjecture previously disclosed by Akbari et al.