ON THE IRREGULARITY STRENGTH AND MODULAR IRREGULARITY STRENGTH OF FRIENDSHIP GRAPHS AND ITS DISJOINT UNION

Total vertex irregularity strength of disjoint union of Helm graphs

A total vertex irregular k-labeling φ of a graph G is a labeling of the vertices and edges of G with labels from the set {1, 2, . . . , k} in such a way that for any two different vertices x and y their weights wt(x) and wt(y) are distinct. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x. The minimum k for which the graph G...

On the Total Irregularity Strength of Disjoint Union of Arbitrary Graphs

Irregularity strength of dense graphs

Let G be a simple graph of order n with no isolated vertices and no isolated edges. For a positive integer w, an assignment f on G is a function f : E(G) → {1, 2, . . . , w}. For a vertex v, f(v) is defined as the sum f(e) over all edges e of G incident with v. f is called irregular, if all f(v) are distinct. The smallest w for which there exists an irregular assignment on G is called the irreg...

Irregularity Strength of Regular Graphs

Let G be a simple graph with no isolated edges and at most one isolated vertex. For a positive integer w, a w-weighting of G is a map f : E(G) → {1, 2, . . . , w}. An irregularity strength of G, s(G), is the smallest w such that there is a w-weighting of G for which ∑

Product irregularity strength of graphs

Consider a simple graph G with no isolated edges and at most one isolated vertex. A labeling w : E(G) → {1, 2, . . . ,m} is called product-irregular, if all product degrees pdG(v) = ∏ e3v w(e) are distinct. The goal is to obtain a product-irregular labeling that minimizes the maximum label. This minimum value is called the product irregularity strength. The analogous concept of irregularity str...