$b$-vectors of chordal graphs

TL;DR

This paper explores the $b$-vector of chordal graphs, revealing its connection to the graph's structural properties, connectivity, clique dominance, and Betti numbers of associated algebraic structures.

Contribution

It establishes new relationships between the $b$-vector and structural features of chordal graphs, including their connectivity and algebraic invariants.

Findings

The $b$-vector encodes connectivity aspects of chordal graphs.

The $b$-vector relates to clique dominance in the graph.

Connections between the $b$-vector and Betti numbers are demonstrated.

Abstract

The $b$-vector $(b_1,b_2\ldots,b_d)$ of a graph $G$ is defined in terms of its clique vector $(c_1,c_2\ldots,c_d)$ by the equation $\sum^d_{i=1}b_i(x+1)^{i-1}=\sum^d_{i=1} c_i x^{i-1},$ 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 $G$.