An inclusive distance vertex irregular labeling of a graph G is an assignment of positive integers \(\{1, 2, \ldots , k\}\) to the vertices of G such that for every vertex the sum of numbers assigned to its closed neighborhood is different. The minimum number k for which exists an inclusive distance vertex irregular labeling of G is denoted by \({\widehat{\mathrm{dis}}}(G)\). In this paper we prove that for a simple graph G on n vertices in which no two vertices have the same closed neighborhood \({\widehat{\mathrm{dis}}}(G)\le n^2\).

中文翻译：

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包含距离顶点不规则强度的上限

图G的包含距离顶点不规则标记是将正整数\(\{1, 2, \ldots , k\}\)分配给G的顶点，使得对于每个顶点分配给其闭合的数字的总和小区不一样。存在G的包含距离顶点不规则标记的最小数k表示为\({\widehat{\mathrm{dis}}}(G)\)。在本文中，我们证明了对于n个顶点上的简单图G，其中没有两个顶点具有相同的闭邻域\({\widehat{\mathrm{dis}}}(G)\le n^2\)。