Omega invariant of graphs and cyclicness

Abstract

Starting with the formula for the number of leaves of a tree, two of the authors recently defined a new graph invariant called Omega denoted by Ω(G) only in terms of a given degree sequence. This invariant is shown to have many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness. Also some extremal problems are recently solved by means of it. In this paper, some new properties of Omega invariant, especially those related to the cyclicness and the number of components of the realized graphs are obtained.