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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
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[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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