Algebraic study on rooted products of graphs and multi-clique corona graphs

TL;DR

This paper explores the algebraic properties of rooted products and multi-clique corona graphs, establishing conditions for Cohen-Macaulayness and providing formulas for algebraic invariants.

Contribution

It introduces 2-Cohen-Macaulayness for edge ideals, characterizes when rooted products are Cohen-Macaulay, and defines multi-clique corona graphs with their algebraic properties.

Findings

Rooted products can be Cohen-Macaulay under specific conditions.

Multi-clique corona graphs are vertex decomposable and sequentially Cohen-Macaulay.

Formulas for projective dimension and Castelnuovo-Mumford regularity are provided.

Abstract

In this paper, we study rooted products of graphs from the perspective of combinatorial commutative algebra. For edge ideals, we introduce the 2-Cohen-Macaulayness with respect to a vertex and use it to investigate when edge ideals of rooted products of graphs are Cohen-Macaulay. Moreover, we completely determine when attaching a graph on at most six vertices to a given graph as rooted products, yields a Cohen-Macaulay edge ideal. Also, we define mulit-clique corona graphs as a generalization of clique-corona graphs and multi-whisker graphs. We prove that multi-clique corona graphs are vertex decomposable and hence sequentially Cohen-Macaulay. Also, we give formulas for the projective dimension and the Castelnuovo-Mumford regularity.