Clar and fries number for benzenoid systems

Magistrsko delo zajema področji organske kemije in teorije grafov. Benzenoidni ogljikovodiki so aromatske spojine, katerih razporeditev π-elektronov lahko prikažemo s pomočjo Kekuléjevih struktur. Matematično gledano so to popolna prirejanja. Šestkotnik benzenoidnega sistema, ki vsebuje natanko tri povezave popolnega prirejanja, imenujemo sekstet. Moč največje množice sekstetov glede na vse Kekuléjeve strukture benzenoidnega sistema imenujemo Friesovo število. Z dodatno zahtevano neodvisnostjo sekstetov dobimo Clarovo število. Glavni izrek magistrske naloge pomaga pri iskanju Clarovega števila velike družine benzenoidnih sistemov in omogoča primerjavo stabilnosti različnih benzenoidnih ogljikovodikov. V prvem delu predstavimo nekaj osnov teorije grafov. Nadalje proučimo kemijsko ozadje benzenoidnih ogljikovodikov in njihovo preoblikovanje v matematični jezik benzenoidnih sistemov. V drugem delu spoznamo Friesovo in Clarovo število ter dokažemo nekaj lem, potrebnih za dokazovanje glavnega izreka. Pokažemo, da obstaja preprost enoličen način iskanja Friesovega števila omejene množice benzenoidnih sistemov. Z glavnim izrekom za izbrano podmnožico benzenoidnih sistemov dokažemo, da je množica sekstetov, ki daje Clarovo število, vsebovana v množici sekstetov, ki ponudi Friesovo število. V zadnjem delu magistrske naloge definiramo pojem stabilnosti benzenoidnega ogljikovodika. Ugotovitve podkrepimo s primeri in teoretična spoznanja primerjamo s praktično dobljenimi vrednostmi. Za konec spoznamo najbolj stabilno podmnožico benzenoidnih ogljikovodikov, popolne benzenoidne ogljikovodike.Master thesis includes fields of organic chemistry and graph theory. Benzenoid hydrocarbons are aromatic compounds for which the arrangement of π-electrons can be represented by the Kekulé structures. Mathematically, these are perfect matchings. A hexagon of a benzenoid system which contains exactly three edges of a perfect matching is called a benzene face. The power of the maximum set of benzene faces with regard to all the Kekulé structures of the benzenoid system is called the Fries number. With an additionally required independency of benzene faces, we get the Clar number. The main theorem presented in the thesis helps to search for the Clar number of a large family of benzenoid systems and enables the comparison of the stabillity of different benzenoid hydrocarbons. In the first part, we introduce some basic notions of graph theory. Furthermore, we study a part of chemical background of benzenoid hydrocarbons and their transformation into the mathematical language of benzenoid systems. Next, we introduce the Fries and Clar numbers and prove a few lemmas which help us prove the main theorem. We point out the existence of a simple and unique way of searching for the Fries number for a limited set of benzenoid systems. For that same set of benzenoid systems we prove main theorem which claims that the set of benzene faces which gives the Clar number is contained in the set of benzene faces, which gives the Fries number. In the last part of the thesis, we define the concept of stability of benzenoid hydrocarbons. We substantiate the theoretical conclusions with examples and compare the results with the experimental values. In the end, we learn about the most stable subset of benzenoid hydrocarbons, the all-benzenoid hydrocarbons