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On graphs having $$\sigma$$-polynomials of the same degree. (English) Zbl 0771.05038
The graphs considered in this paper are finite and undirected, with no loops or parallel edges. The $$\sigma$$-polynomial $$\sigma(G,t)$$ of a graph $$G$$ with $$p$$ vertices is defined in R. R. Korfhage [$$\sigma$$- polynomials and graph coloring, J. Comb. Theory, Ser. B 24, No. 2, 137- 153 (1978)] as follows: if the chromatic polynomial $$p(G,\lambda)$$ of $$G$$ is $$\sum^{p-\chi(G)}_{i=0}a_ i\lambda(\lambda-1)\cdots(\lambda-(p- i)+1)$$, where $$\chi(G)$$ is the chromatic number of $$G$$, then $$\sigma(G,t)=\sum^{p-\chi(G)}_{i=0}a_ it^{p-\chi(G)-i}$$. A theorem in R. C. Read [An introduction to chromatic polynomials, J. Comb. Theory 4, 52-71 (1967; Zbl 0173.262)] which identifies $$a_ i$$ as the number of subgraphs of the complement of $$G$$ which are isomorphic to the union of complete graphs with a total of $$i$$ vertices, is used to obtain a necessary and sufficient condition for the degree $$p-\chi(G)$$ of $$\sigma(G,t)$$ to be $$k$$ for any positive integer $$k$$. This generalizes the condition found in the above-mentioned paper of R. R. Korfhage for $$k=0$$ and 1 and in M. Dhurandhar [J. Comb. Theory, Ser. B 37, 210- 220 (1984; Zbl 0554.05030)]. This condition is then used to construct all the graphs whose $$\sigma$$-polynomials are of degree 2, 3 and 4; the results for degree 2 agree with those in R. W. Frucht and R. E. Giudici [Ars. Comb. 16-A, 161-172 (1983; Zbl 0536.05026)].

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
$$\sigma$$-polynomial; chromatic number
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##### References:
 [1] Dhurandhar, M., Characterization of quadratic and cubic σ-polynomials, J. combin. theory ser. B, 37, 210-220, (1984) · Zbl 0554.05030 [2] Frucht, R.W.; Giudici, R.E., Some chromatically unique graphs with seven points, Ars combin., 16-A, 161-172, (1983) · Zbl 0536.05026 [3] Harary, F., Graph theory, (1969), Addison-Wesley Reading, MA · Zbl 0797.05064 [4] Korfhage, R.R., σ-polynomials and graph coloring, J. combin. theory ser. B, 24, 137-153, (1978) · Zbl 0845.05043 [5] Li, N.-Z.; Whitehead, E.G., Graph theory and its applications: east and west, Ann. N.Y. acad. sci., 576, 328-335, (1989), Classification of graphs having cubic σ-polynomials [6] Read, R.C., An introduction to chromatic polynomials, J. combin. theory, 4, 52-71, (1968) · Zbl 0173.26203
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