It is shown that a geometric realization of the clique complex of a connected chordal graph is homologically trivial and as a consequence of this it is always the case for any connected chordal graph G that âˆ‘_(k=1)^Ï‰(G)â–’(-1)^(k-1) Î·_k (G)=1, where Î·_k (G) is the number of cliques of order k in G and Ï‰(G) is the clique number of G.
algebraic graph theory, chordal graph, clique complex, hypergraph, homology, Mayer-Vietoris theorem, graph invariant, Euler-PoincarÃ© formula
- McKee, T.; McMorris, F. Topics in Intersection Graph Theory; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 199
- Roberts, F. Discrete Mathematical Models with Application to Social, Biological, and Environmental Problems; Prentice-Hall, Inc.: Englewood Cliffs, NJ, USA, 1976
- Brigham, R.; Dutton, R. A compilation of relations between graph invariants. Networks 1985, 15, pp. 73-107.
- Beeri, C.; Fagin, R.; Maier, D.; Yannakakis, M. On the Desirability of Acyclic Database Schemes. Journal of the Association for Computing Machinery 1983, 30, pp. 479-513
- Fagin, R. Degrees of Acyclicity for Hypergraphs and Relational Database Schemes. Journal of the Association for Computing Machinery 1983, 30, pp. 514-550
- Hocking, J.; Young, G. Topology; Addison-Wesley: Reading, MA, USA, 1961; p. 242
- Munkres, J. Elements of Algebraic Topology; Addison-Wesley: Reading, MA, USA, 1984; p. 142
- Rotman, J. An Introduction to the Theory of Groups; Allyn and Bacon, Boston, MA, USA, 1984; p. 254
- Dâ€™Atri, A.; Moscarini, M. On Hypergraph Acyclicity and Graph Chordality. Information Processing Letters 1988, 29, pp.271-274
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