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Given a clique complex $K$ constructed from a discrete set of vertices (i.e. its faces are isomorphic to the set of cliques in the 1-skeleton of $K$.), it seems that the Betti numbers $\beta_k$ defined for the homology of $K$ are in some sense correlated since the $n(0\leq n\leq\dim K)$-simplexes in $K$ are completely determined by the 1-skeleton. Beyond the classical result given by discrete Morse theory

$$-f_{k-1}+f_{k}-f_{k+1}\leq \,\beta_k \, \leq f_k$$

where $f_k$ are the number of $k$-faces in $K$, are there further results concerning the relations between $\beta_k$ and $\beta_{k+1}$ ? Since there exists many upper bounds for the sum of Betti numbers for a given clique complex (e.g. here), is such bound providing any information about the relations in between any two Betti numbers? Or it is just casual counting bound?

Any comment or reference is highly appreciated.

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    $\begingroup$ The barycentric subdivision of a simplicity complex is always a flag complex and this doesn't change the homology. $\endgroup$
    – Benjamin Steinberg
    Commented Feb 26, 2018 at 18:34
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    $\begingroup$ Since you ask "are there further...relations between $\beta_k$ and $\beta_{k+1}$, and since the inequality you are citing holds for any simplicial complex (not only for clique complexes of the 1-skeleton), it is not off-topic to mention the usual formula, also valid in general and not specific to your setting, for the Euler-characteristic: $f_0-f_1+\dotsm+(-1)^{\mathrm{dim}(K)} f_{\mathrm{dim}(K)} = \beta_0 - \beta_1 + \dotsm + (-1)^{\mathrm{dim}(K)}\beta_{\mathrm{dim}(K)}$. Again, this is a validity on the class of all abstract simplicial complexes, not specific to your question. $\endgroup$
    – Peter Heinig
    Commented Feb 26, 2018 at 19:13
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    $\begingroup$ @PeterHeinig Thanks! Sure, but for a more general simplicial complex I am not even sure in that case if two consecutive Betti numbers are related since there could be much more possibilities than a clique complex. $\endgroup$
    – Henry.L
    Commented Feb 26, 2018 at 19:54

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