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Chromatic polynomials of connected graphs. (English) Zbl 0535.05053
We divide the family of connected graphs with $$n(\geq 3)$$ vertices into 3 mutually disjoint subfamilies, namely, $${\mathcal F}_ 1$$ consists of all connected graphs each of which has at least one cutpoint, $${\mathcal F}_ 2$$ consists of all 2-connected graphs each of which has no subgraph homeomorphic to the complete graph $$K_ 4$$ with 4 vertices, and $${\mathcal F}_ 3$$ consists of all 2-connected graphs each of which has at least one subgraph homeomorphic to $$K_ 4$$. Let G be a connected graph with $$n(\geq 3)$$ vertices, and $$P(G,\lambda)=\lambda^ n-a_{n- 1}\lambda^{n-1}+...\pm a_ 1\lambda$$ be its chromatic polynomial. Replacing $$\lambda$$ in P(G,$$\lambda)$$ by $$\omega +1$$, we have $$P(G,\lambda)=Q(G,\omega)=\omega^ n+b_{n-1}\omega^{n-1}+...+b_ 1\omega.$$ Here, we show that (1) $$G\in {\mathcal F}_ 1$$ if and only if $$| b_ 1| =0$$, (2) $$G\in {\mathcal F}_ 2$$ if and only if $$| b_ 1| =1$$, and (3) $$G\in {\mathcal F}_ 3$$ if and only if $$| b_ 1| \geq 2$$.

##### MSC:
 05C99 Graph theory 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C15 Coloring of graphs and hypergraphs 05C40 Connectivity
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##### References:
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