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Classification of chromatically unique graphs having quadratic $$\sigma$$- polynomials. (English) Zbl 0686.05021
Let P(G) denote the chromatic polynomial of a finite simple graph. The graph G is called chromatically unique if $$P(G)=P(H)$$ implies that G is isomorphic with H. Chromatically unique graphs have been studied by C.-Y. Chao and the second author [Lect. Notes Math. 642, 121-131 (1978; Zbl 0369.05032)].
Let the chromatic polynomial of a graph G with p vertices be given in terms of the factorial basis $$\{(\lambda)_ i:$$ $$i=0,1,...\}$$, where $$(\lambda)_ i=\lambda (\lambda -1)...(\lambda -i+1)$$ (i$$\geq 1)$$, $$(\lambda)_ 0=1$$, and let $$\chi$$ (G) denote the chromatic number of G. Then there are integers $$a_ 0,a_ 1,...,a_ h$$ with $$P(G)=\sum^{h}_{i=1}a_ i(\lambda)_{p-i}$$ where $$h=p-\chi (G).$$ Now the $$\sigma$$-polynomial of G is defined by $$\sigma (G)=\sum^{h}_{i=0}a_ i\sigma^{h-i}$$. R. W. Frucht and R. E. Giudici [Ars Comb. 16-A, 161-172 (1983; Zbl 0536.05026)] classified all graphs having quadratic $$\sigma$$-polynomials.
In the present paper such a characterization is given for all chromatically unique graphs having quadratic $$\sigma$$-polynomials.
Reviewer: U.Baumann

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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##### References:
 [1] and , On chromatic equivalence of graphs. in Theory and Applications of Graphs. and , Eds., Springer-Verlag Lecture Notes in Mathematics, Vol. 642, Springer-Verlag, New York (1978). · Zbl 0369.05032 [2] Frucht, Ars Combinatoria 16-A pp 161– (1983) [3] Korfhage, J. Combinatorial Theory, Series B 24 pp 137– (1978) [4] and , Characterization of graphs having quadratic ??-polynomials, to appear. [5] Read, J. Combinatorial Theory 4 pp 52– (1968) · Zbl 0165.32802
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