Clique Vectors of $k$-Connected Chordal Graphs

TL;DR

This paper characterizes all possible clique vectors of $k$-connected chordal graphs using tools from commutative algebra, providing a comprehensive understanding of their clique structure.

Contribution

It introduces a complete algebraic characterization of clique vectors specifically for $k$-connected chordal graphs, a class of graphs with important combinatorial properties.

Findings

Provides a complete characterization of clique vectors for $k$-connected chordal graphs

Uses commutative algebra tools to analyze graph clique structures

Advances understanding of the relationship between connectivity and clique distributions

Abstract

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.