On inclusive distance vertex irregular labelings

<p>For a simple graph G, a vertex labeling f:V(G)\to {1, 2, ..., k} is called a <em>k-labeling</em>. The weight of a vertex v, denoted by $wt_f(v)$ is the sum of all vertex labels of vertices in the closed neighborhood of the vertex v. A vertex k-labeling is defined to be an <em>inclusive distance vertex irregular distance k-labeling</em> of G if for every two different vertices u and v there is $wt_f(u)\ne wt_f(v)$. The minimum k for which the graph G has a vertex irregular distance k-labeling is called the <em>inclusive distance vertex irregularity strength</em> of G. In this paper we establish a lower bound of the inclusive distance vertex irregularity strength for any graph and determine the exact value of this parameter for several families of graphs.