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On a Class of Polynomials Associated with the Stars of a Graph and its Application to Node-Disjoint Decompositions of Complete Graphs and Complete Bipartite Graphs into Stars

Published online by Cambridge University Press:  20 November 2018

E. J. Farrell
E. J. Farrell*
Affiliation:
Dept. of Mathematics, University of the West Indies, St. Augustine, Trinidad, W.I.
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Abstract

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A star is a connected graph in which every node but possibly one has valency 1. Let G be a graph and C a spanning subgraph of G in which every component is a star. With each component α of C let us associate a weight wα. Let Пα wα be the weight associated with the entire subgraph G the star polynomial of G is ΣПα wα where the summation is taken over all spanning subgraphs of G consisting of stars. In this paper an algorithm for finding star polynomials of graphs is given. The star polynomials of various classes of graphs are then found, and some results about node-disjoint decomposition of complete graphs and complete bipartite graphs are deduced.

Keywords

05C1005C99Graphstarstar polynomialdecomposition of a graphgenerating functionnode disjoint subgraphs
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 1 , 01 March 1979 , pp. 35 - 46
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Ae, Tadashi, Yamamoto, Seigo, Yoshida, Noriyosha, Line-disjoint Decomposition of Complete Graphs into stars, J. Comb. Theory Ser. B (to appear)Google Scholar
2. Cain, Pauline Decomposition of Complete Graphs into Stars, Bull. Austral. Math. Soc. Vol. 10 (1974), 23-30.Google Scholar
3. Farrell, E. J., On a General Class of Graph Polynomials J. Comb. Theory Ser. B (to appear).Google Scholar
4. Harary, F., Graph Theory, Addison-Wesley Pub. Co. Inc., Reading, Mass. (1969).Google Scholar