Let G = (V,E), V = {1,2, . . . ,n}, be a simple graph without isolated vertices, with vertex degree sequence d1 ≥ d2 ≥ ·· · ≥ dn > 0, di = d(i). A graph G is regular if and only if d1 = d2 = · · · = dn. A graph invariant I(G) is measure of irregularity of graph G with the property I(G) = 0 if and only ifG is regular, and I(G)> 0 otherwise. In this paper we introduce some new irregularity measures.

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 ...

Let is a simple and connected graph with as vertex set edge set. Vertex labeling on inclusive local irregularity coloring defined by mapping the function of . In other words, an so that its weight value obtained adding up labels neighboring label. The chromatic number minimum colors from in G, denoted this paper, we learn about determine book graphs.

For any graph G, let ni be the number of vertices of degree i, and λ(G) = maxi≤j{ ni+···+nj+i−1 j }. This is a general lower bound on the irregularity strength of graph G. All known facts suggest that for connected graphs, this is the actual irregularity strength up to an additive constant. In fact, this was conjectured to be the truth for regular graphs and for trees. Here we find an infinite ...

An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we take an iterative approach to calculating the irregularity stren...