The line graph of a tree and its edge ideal

TL;DR

This paper characterizes trees whose line graphs have edge ideals with linear resolutions, explores their properties, and computes various algebraic and graph invariants for these structures.

Contribution

It provides a complete characterization of trees with line graphs having co-chordal properties and computes key algebraic invariants for their edge ideals.

Findings

Characterization of trees with line graphs having linear resolution edge ideals

Calculation of the second Betti number of the edge ideal of line graphs

Determination of the number of cycles in the complement of line graphs

Abstract

We describe all the trees with the property that the corresponding edge ideal of their line graph has a linear resolution. As a consequence, we give a complete characterization of those trees $T$ for which the line graph $L(T)$ is co-chordal. We also compute the second Betti number of the edge ideal of $L(T)$ and we determine the number of cycles in $\overline{L(T)}$. As a consequence, we obtain also the first Zagreb index of a graph. For edge ideals of line graphs of caterpillar graphs we determine the Krull dimension, the Castelnuovo-Mumford regularity, and the projective dimension under some additional assumption on the degrees of the cutpoints.