The total irregularity of a graph G is defined as irrt .G/ D 1 2 P u;v2V.G/ jdG.u/ dG.v/j, where dG.u/ denotes the degree of a vertex u 2 V.G/. In this paper we give (sharp) upper bounds on the total irregularity of graphs under several graph operations including join, lexicographic product, Cartesian product, strong product, direct product, corona product, disjunction and symmetric difference....

In his paper Linear equations in non-commutative fields (Ann. of Math. vol. 32 (1931) pp. 463-477) Professor Oystein Ore defines regularity and irregularity of rings and an "order of irregularity" in such a way that regularity becomes irregularity of order 1. Herewith the following problem is proposed: Do irregular rings of order n>\ really exist? If so of what type are they? In this note these...

This paper proposes a nonparametric steganalysis method for quantization index modulation (QIM) based steganography. The proposed steganalysis method uses irregularity (or randomness) in the test-image to distinguish between the coverand the stego-image. We have shown that plain-quantization (quantization without message embedding) induces regularity in the resulting quantized-image; whereas me...

Let Γ(G) be a set of all simple graphs of order n and sizem, without isolated vertices, with vertex degree sequence d1 ≥ d2 ≥ ·· · ≥ dn > 0. A graph G is regular if and only if d1 = d2 = · · ·= dn. Each mapping Irr : Γ(G) 7→ [0,+∞) with the property Irr(G) = 0 if and only if G is regular, is referred to as irregularity measure of graph. In this paper we introduce some new irregularity measures ...