On \(\sigma\)-polynomials and a class of chromatically unique graphs.

*(English)*Zbl 0774.05039The \(\sigma\)-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.

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.

Reviewer: T.R.Walsh (Montreal)

##### Keywords:

chromatically unique graphs; chromatic number; chromatic polynomial; upper bound; symmetric functions
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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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