On the vertex degree indices of connected graphs Tomislav Doˇ sli´ c 1 ,Tam´asR´ eti 2 , Damir Vukiˇ cevi´ c 3 1 Faculty of Civil Engineering, University of Zagreb, Kaˇ ci´ ceva 26, Zagreb, Croatia e-mail: doslic@grad.hr 2 Sz´ echenyi Istv´ an University, Egyetem t´ er 1, 9026 Gy¨ or, Hungary e-mail: reti@sze.hu 3 Department of Mathematics, University of Split, N. Tesle 12, 21000 Split, Croatia e-mail: vukicevi@pmfst.hr Abstract We introduce a family of invariants defined in terms of positive functions of degrees of vertices in a graph. A member of the family that measures the average degree of neighbors of vertices in a graph is then investigated for the predictive potential for stability in the class of generalized fullerenes. Keywords: generalized fullerene, average neighbor degree number 1 Introduction One of the cornerstones of mathematical chemistry and in particular chemical graph theory is the belief that various physico-chemical properties of molecules can be deduced from some invariants of the corresponding graphs. The belief is based on the fact that the properties are determined by the types of atoms combined in the molecule and by their relative positions. Since both concepts can be well modeled in the language of graphs, it is natural to expect that there will be non-trivial correlations between physico-chemical properties and some suitably chosen graph-theoretical invariants. A graph-theoretical invariant, more commonly called a topological index, is a real number derived from the structure of a graph in such a way that it does not depend on the labeling 1