NEW CONCEPTS AND RESULTS ON THE AVERAGE DEGREE OF A GRAPH

The idea of equilibrium of a graph G, initially applied to maximal outerplanar graphs (mops), was conceived to observe how the vertex degree distribution affects the average degree of the graph, d(G). In this work, we formally extend the concept to graphs in general. From d(G), two new parameters are introduced - the top and the gap of G, sustaining the definitions of tuner set, balanced and non-balanced graphs. We show properties of the new concepts when applied to particular families of graphs as trees and unicyclic graphs.We also establish bounds to the top of non-balanced graphs with integer average degree and we characterize their tuner sets.