$b$-vectors of chordal graphs

The $b$-vector $(b_1,b_2\dots ,b_d)$ of a graph $G$ is defined in terms of its clique vector $(c_1,c_2\dots ,c_d)$ by the equation ${\sum }_{i=1}^d{b}_i(x+1{)}^{i-1}={\sum }_{i=1}^d{c}_i{x}^{i-1},$ where $d$ is the largest cardinality of a clique in $G$. We study the relation of the $b$-vector of a chordal graph $G$ with some structural properties of $G$. In particular, we show that the $b$-vector encodes different aspects of the connectivity and clique dominance of $G$. Furthermore, we relate the $b$-vector with the Betti numbers of the Stanley-Reisner ring associated to clique simplicial complex of $G$.