Local Cohomology Modules of Binomial Edge Ideals of Complements of Connected Graphs of Girth at Least 5

We calculate the local cohomology modules of the binomial edge ideals of the complements of connected graphs of girth at least 5 using the tools introduced by \`Alvarez Montaner in arXiv:1901.08645. We then use this calculation to compute the depth, dimension, and regularity of these binomial edge ideals.
Comment: 15 pages, v2 includes small changes to several proofs for clarity, and reorganises section 2 into subsections (whilst some numbering has changed, no new results are introduced)