Some Distance and Degree Graph INVARIANTS AND FULLERENE STRUCTURES

In the thesis we concentrate to the part of graph theory that can be applied in chemistry. One of the aims is applying graph-theoretical methods to predict the properties of a chemi-cal compound based on its molecule structure. Molecular descriptors or topological indices is one way of predicting some properties. We dedicate our attention to Zagreb indices, a modification of Randic ́ index called R ′ index, and Gutman index. For a simple graphG with n vertices andm edges, the inequalityM1(G)/n ≤M2(G)/m, where M1(G) and M2(G) are the first and the second Zagreb indices of G, is known as Zagreb indices inequality. We characterize the intervals of vertex degrees that satisfy this inequality, and find an infinite family of connected graphs dissatisfying this inequality. We also present an algorithm that decides if an arbitrary set of vertex degrees satisfies the inequality and consider variable Zagreb index inequality. We also determine the graphs with extremal values for Gutman and R ′ indices, and find a triangle-free graph on n vertices with minimum R′(G) index. Fullerene molecules are carbon cage molecules arranged only in pentagons and hexagons