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The inverse eigenvalue problem of a graph $G$ is the problem of characterizing all lists of eigenvalues of real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of $G$. The strong spectral property is a powerful tool in this problem, which identifies matrices whose entries can be perturbed while controlling the pattern and preserving the eigenvalues. The Matrix Liberation Lemma introduced by Barrett et al.~in 2020 advances the notion to a more general setting. In this paper we revisit the Matrix Liberation Lemma and prove an equivalent statement, that reduces some of the technical difficulties in applying the result. We test our method on matrices of the form $M=A \oplus B$ and show how this new approach supplements the results that can be obtained from the strong spectral property only. While extending this notion to the direct sums of graphs, we discover a surprising connection with the zero forcing game on Cartesian products of graphs. Throughout the paper we apply our results to resolve a selection of open cases for the inverse eigenvalue problem of a graph on six vertices.

Rational functions in $d$ variables over a field ${mathbb F}$ are actual (partial) functions from ${mathbb F}^d$ to ${mathbb F}$ that can be formed using coordinate functions and rational operations (addition, scalar multiplication, multiplication, inversion). Such functions form a field. Noncommutative rational functions in $d$ variables over ${mathbb F}$ are partial functions from $d$-tuples of equally sized square matrices over ${mathbb F}$ to matrices of the same size that can be formed using coordinate functions and rational operations. Such functions form a skew-field where every relation between the variables follows from the existence of inverses of nonzero elements, hence, the skew-field of noncommutative rational functions is also called a free skew-field. One of the major problems of invariant theory is Noether’s problem – given an action of a finite group on a field of rational functions, is the field of invariant functions isomorphic to a field of rational functions? In the thesis, we investigate a noncommutative version of Noether’s problem – given an action of a finite group on a free skew-field, is the skew-field of invariant functions free, i.e., isomorphic to a free skew-field? We study the actions of finite abelian groups on the free skew-field over ${mathbb C}$ and ${mathbb R}$ that are given by linear representations and show that their invariant skew-subfields are always free. We define complete representations – a type of linear representation of solvable groups that admit an inductive extension of the result for linear actions of abelian groups. For example, the standard representations of the symmetric groups $S_3$ and $S_4$ are complete. We also investigate so-called multiplicative actions of finite cyclic groups – actions that are defined by an automorphism of a free group and show that they are equivalent to linear actions. We give some interesting examples of invariants of cyclic groups over ${mathbb Q}$ and invariants of the general linear group. The last part of the thesis is more group theoretic. We give an alternative characterisation of the groups that admit complete representations and name them totally pseudo-unramified groups. We present some group theoretic properties of totally pseudo-unramified groups and classify totally pseudo-unramified $p$-groups of rank up to five. Racionalne funkcije v $d$ spremenljivkah nad poljem ${mathbb F}$ so delne funkcije iz ${mathbb F}^d$ v ${mathbb F}$, ki jih lahko izrazimo s koordinatnimi funkcijami (spremenljivkami) in racionalnimi operacijami (seštevanje, množenje, deljenje). Nekomutativne racionalne funkcije v $d$ spremenljivkah nad ${mathbb F}$ so delne funkcije, ki slikajo iz $d$-teric kvadratnih matrik iste velikosti v kvadratne matrike, ki jih lahko izrazimo s koordinatnimi funkcijami in racionalnimi operacijami. Take funkcije tvorijo obseg, v katerem je vsaka relacija med spremenljivkami posledica obstoja inverzov neničelnih elementov, zato obseg nekomutativnih racionalnih funkcij imenujemo tudi prosti obseg. Eden glavnih problemov teorije invariant je problem Emmy Noether – ali je, za dano delovanje končne grupe na prosto polje, polje invariant izomorfno prostemu polju? V disertaciji obravnavamo nekomutativno različico problema Emmy Noether – ali je, za dano delovanje končne grupe na prost obseg, obseg invariant izomorfen prostemu obsegu? Preučujemo delovanja končnih abelovih grup na proste obsege nad ${mathbb C}$ in ${mathbb R}$, ki so določena z linearnimi upodobitvami. Pokažemo, da je obseg njihovih invariant vedno prost. Definiramo kompletne upodobitve – družino linearnih upodobitev rešljivih grup, ki omogočajo induktivno razširitev rezultata o delovanjih abelovih grup. Primera kompletnih upodobitev sta standardni upodobitvi simetričnih grup $S_3$ in $S_4$. Obravnavamo tudi multiplikativna delovanja končnih cikličnih grup – delovanja, ki so določena z avtomorfizmom proste grupe. Predstavimo nekaj zanimivih primerov invariant cikličnih grup nad ${mathbb Q}$ in invariant splošne linearne grupe. V zadnjem delu disertacije obravnavamo grupe, ki imajo kompletne upodobitve – imenujemo jih popolnoma psevdo-nerazvejane grupe. Predstavimo lastnosti popolnoma psevdo-nerazvejanih grup in karakteriziramo popolnoma psevdo-nerazvejane $p$-grupe ranga največ pet.