A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums labels taken over distinct hyperedges are distinct. Let s(H) be smallest integer N such that there sum-distinguishing H with each label at most N. The largest value all hypergraphs on n vertices m denoted s(n,m). We prove s(n,m) almost-quadratic in as long not too large. More precisely, following h...

The irregularity of a graph G is defined as irr(G) = ∑ |d(x) − d(y)| where d(x) is the degree of vertex x and the summation embraces all pairs of adjacent vertices of G. We characterize the graphs minimum and maximum values of irr.

A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...

in this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. also, density and balanced irregular vague graphs are discussed and some of their properties are established. finally we give an application of vague digraphs.