A Kruskal–Katona type theorem for graphs

AbstractA bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal–Katona theorem. A bound on non-consecutive clique numbers is also proven

Clique Vectors of kk-Connected Chordal Graphs

The clique vector $\mathfrak{c}(G)$ of a graph $G$ is the sequence $(c_1, c_2, \ldots,c_d)$ in $\mathbb{N}^d$, where $c_i$ is the number of cliques in $G$ with $i$ vertices and $d$ is the largest cardinality of a clique in $G$. In this note, we use tools from commutative algebra to characterize all possible clique vectors of $k$-connected chordal graphs