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For a graph \(G\), assign an integer value weight to each edge. For a vertex \(v\), the label of v is the sum of weights of the edges incident with it. Further, the weighting is irregular if all the vertex labels are distinct. It is well known that if \(G\) has at most one isolated vertex and no isolated edges, then there exist irregular assignments, in fact, using positive edge weights.
In this paper, we consider the following special weighting:
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– If \(G\) has order \( 2 k + 1\), then a consecutive labeling is an assignment where the vertex labels are precisely \(-k, -k+1, \ldots, -1, 0, 1, 2, \ldots, k-1, k\).
– If \(G\) has order \( 2k\), then a consecutive labeling is an assignment where the vertex labels are precisely \( -k+1, \ldots, -1, 0, 0, 1, 2, \ldots, k-1\).
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Here we show that every graph which has an irregular assignment also has a consecutive labeling. This concept is extended by considering all consecutive labelings and looking for one that has the smallest maximum, in absolute value, edge weight. This weight is referred to as the consecutive strength. Results parallel to the concept of irregularity strength are presented.