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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:
 [1] J.A. Bondy and U.S.R. Murty, Graph Theory with Application (Elsevier. North-Holland, New York). · Zbl 1134.05001 [2] N. Biggs, Algebraic Graph Theory (Cambridge Univ. Press, Cambridge). · Zbl 0284.05101 [3] Korfhage, R.R., Σ-polynomials and graph colouring, J. combin. theory ser. B, 24, 137-153, (1978) · Zbl 0845.05043 [4] Frucht, R.W.; Giudici, R.E., Some chromatically unique graphs with seven points, Ars combin., 16A, 161-172, (1983) · Zbl 0536.05026 [5] Read, R.C., An introduction of chromatic polynomials, J. combin. theory, 4, 52-71, (1968) · Zbl 0173.26203 [6] Dhurandhar, M., Characterization of quadratic and cube σ-polynomials, J. combin. theory ser. B, 37, 210-220, (1984) · Zbl 0554.05030 [7] Xu, S.-J., On σ-polynomials, Discrete math., 69, 189-194, (1988) · Zbl 0658.05025
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