For a simple, undirected graph G with, at most one isolated vertex and no edges, labeling f:E(G)→{1,2,…,k1} of positive integers to the edges is called irregular if weights each has different value. The integer k1 then irregularity strength G. If number vertices in or order |G|, μ:E(G)→{1,2,…,k2} modular remainder divided by |G| k2 disjoint union two more graphs, denoted ‘+’, an operation where...

The study of graphs and networks accomplished by topological measures plays an applicable task to obtain their hidden topologies. This procedure has been greatly used in cheminformatics, bioinformatics, biomedicine, where estimations based on graph invariants have made available for effectively communicating with the different challenging tasks. Irregularity are mostly characterization nonregul...

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to local inclusive (respectively, non-inclusive) d-distance irregular of graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight x∈V(G) sum all labels whose distance from x at most d but least 1). The minimum k which there exists called irregularity strength G. In this paper, we p...

The irregularity strength of a simple graph is the smallest integer k for which there exists a weighting of the edges with positive integers at most k such that all the weighted degrees of the vertices are distinct. In this paper we study the irregularity strength of circulant graphs of degree 4. We find the exact value of the strength for a large family of circulant graphs. © 2005 Elsevier B.V...