A survey of the maximal and the minimal nullity in terms of omega invariant on graphs

Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the number of connected components of G, respectively. Wang and Wong in [18] provided formulae for the upper and the lower bounds of the nullity η(G) of G as η(G) = n − 2ν(G) + 2c(G) and η(G) = n − 2ν(G) − c(G), respectively. In this paper, we restate the upper and the lower bounds of nullity η(G) of G utilizing omega invariant and inherently vertex degrees of G. Also, in the case of the maximal and the minimal nullity conditions are satisfied for G, we present both two main inequalities and many inequalities in terms of Omega invariant, analogously cyclomatic number, number of connected components and vertex degrees of G.