Local Inclusive Distance Vertex Irregular Graphs

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 present several basic results on d=1 and determine precise values corresponding invariant certain families graphs.