$t$-clique ideal and $t$-independence ideal of a graph

TL;DR

This paper introduces and analyzes algebraic ideals associated with graphs, specifically clique and independence ideals, exploring their resolutions, Cohen-Macaulay properties, and homological invariants for various graph classes.

Contribution

It characterizes families of clique ideals with linear resolutions and identifies conditions for Cohen-Macaulay quotient rings for certain graph classes.

Findings

Characterization of clique ideals with linear resolutions

Identification of Cohen-Macaulay quotient rings for specific graphs

Homological invariants for clique and independence ideals of path, cycle, and chordal graphs

Abstract

In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized. Moreover some families of graphs for which the quotient ring of their clique ideal is Cohen-Macaulay are introduced and some homological invariants of the clique ideal of a graph $G$ which is the complement of a path graph or a cycle graph, are obtained. Also some algebraic properties of the independence ideal of path graphs, cycle graphs and chordal graphs are studied.