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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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