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
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