Let G = (V,E), V = {v1, v2, . . . , vn}, be a simple connected graph of order n and size m, with vertex degree sequence d1 ≥ d2 ≥ · · · ≥ dn. A graph G is said to be regular if d1 = d2 = · · · = dn. Otherwise it is irregular. In many applications and problems it is important to know how irregular a given graph is. A quantity called degree deviation S(G) = ∑n i=1 ∣∣∣di − 2mn ∣∣∣ can be used as an irregularity measure. Some new lower bounds for S(G) are obtained. A simple formula for computing S(G) for connected bidegreed graphs is derived also. Besides, two novel irregularity measures are introduced