Abstract
The coefficient of the chromatic polynomial counts the number of partitions of the vertex set of a simple and finite graph G into k independent vertex sets, equivalently, it gives the number of proper colorings of G with exactly k colors subject to some constraints. In this work, we study this invariant, we establish new formulas in this context for some families of graphs and we treat some specific cases as Thorn graphs. Consequently, we derive identities for the classical Stirling numbers of the second kind, besides that, this gives rise to new explicit formulae for the r-Stirling numbers of the second kind.
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