It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means no order at least two is, in this way, irregular. However, multigraph can be Chartrand et al., 1988, posed following problem: loopless multigraph, how one determine fewest parallel edges required to ensure all degrees? problem known as labeling and, for its solution, al. introduced irregu...

Assign positive integer weights to the edges of a simple graph with no component isomorphic to Ki or 1£2, in such a way that the graph becomes irregular, i.e., the weight sums at the vertices become pairwise distinct. The minimum of the largest weights assigned over all such irregular assignments on the vertex-disjoint union of complete graphs is determined. The method of proof also yields the ...

We consider undirected graphs without loops or multiple edges. A weighting of a graph G is an assignment of a positive integer w( e) to each edge of G. For a vertex x€V(G), the (weighted) degree d(x) is the sum ofweights on the edges ofG incident to x. The irregularity strength s( G) of a graph G was introduced by Chartrand et al. in [1] a.s the minimum integer t such that G has a weighting wit...

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 weight, minimized over all irregular assignments. In this study, we show that s(G) c1 n / , for graphs with maximum degree n and minimum

We propose two alternative measures of the local irregularity of a graph in terms of its vertex degrees and relate these measures to the order and the global irregularity of the graph measured by the difference of its maximum and minimum vertex degree.

Let G be a graph with p vertices and q edges an injective function, where k is positive integer. If the induced edge labeling defined by for each bijection, then f called odd Fibonacci irregular of G. A which admits graph. The irregularity strength ofes(G) minimum labeling. In this paper, some subdivision graphs obtained from vertex identification determined.

It is an elementary exercise to show that any non-trivial simple graph has two vertices with the same degree. This is not the case for digraphs and multigraphs. We consider generating irregular digraphs from arbitrary digraphs by adding multiple arcs. To this end, we define an irregular labeling of a digraph D to be an arc labeling of the digraph such that the ordered pairs of the sums of the i...

Let Γ=(VΓ,EΓ) be a simple undirected graph with finite vertex set VΓ and edge EΓ. A total n-labeling α:VΓ∪EΓ→{1,2,…,n} is called irregular labeling on Γ if for any two different edges xy x′y′ in EΓ the numbers α(x)+α(xy)+α(y) α(x′)+α(x′y′)+α(y′) are distinct. The smallest positive integer n such that can labeled by irregularity strength of Γ. In this paper, we provide some asymmetric graphs sym...