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On $$\sigma$$-polynomials and a class of chromatically unique graphs. (English) Zbl 0774.05039
The $$\sigma$$-polynomial of a graph was defined in $$(*)$$ R. R. Korfhage [$$\sigma$$-polynomials and graph colouring, J. Comb. Theory, Ser. B 24, No. 2, 137-153 (1978)]. For a graph $$G$$ with $$n$$ vertices and chromatic number $$\chi(G)$$, let $$k=n-\chi(G)$$. If $$P(G,\lambda)$$, the chromatic polynomial of $$G$$, is $$\sum^ k_{i=0}a_ i\lambda(\lambda- 1)\cdots(\lambda-n+i+1)$$, then $$\sigma(G)=\sum^ k_{i=0}a_ i\sigma^{k-i}$$. It is known that $$a_ i>0$$ for $$0\leq i\leq k$$, $$a_ 0=1$$ and $$a_ 1$$ is the number of nonadjacent pairs of vertices of $$G$$. It was shown in $$(*)$$ that $$a_ i\leq{a_ 1\choose i}$$ for $$2\leq i\leq k$$ and in S.-J. Xu [On $$\sigma$$-polynomials, Discrete Math. 69, No. 2, 189-194 (1988; Zbl 0658.05025)] that if $$\sigma^ 2+a\sigma+b$$ is a $$\sigma$$-polynomial, then $$b\leq a^ 2/4$$.
In the paper under review, these results, together with a necessary and sufficient condition for the inequalities in $$(*)$$ to be sharp for at least one value of $$i$$, are obtained as special cases of an upper bound for $$a_ i$$ in terms of elementary symmetric functions.
A graph $$G$$ is called chromatically unique if only isomorphic copies of $$G$$ have the same chromatic polynomial as $$G$$, and a 2-parameter family of chromatically unique graphs is presented.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05E05 Symmetric functions and generalizations
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##### References:
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