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.

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.

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 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 $K$, is 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.

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.