Lynn:
You don't mention the religious denomination for Matthew (Mike) Graham. Roman Catholic or Church of Ireland? The subscription site Roots Ireland has records for both the Loughrea RC parish and the Loughrea C of I parish back before 1824 so possibly he was from a nearby parish which does have records back to the 1820s.
Have you considered DNA testing? There is a East Galway Genealogy and DNA Facebook group which may be of interest.
https://www.facebook.com/groups/555977884943692
There was a Graham family in Loughrea in the 1901 census. The mother was born in Galway but the children were born in Dublin city. The mother died between 1901 and 1911 and the 1911 census shows just the three unmarried children.
1901 census http://www.census.nationalarchives.ie/pages/1901/Galway/Loughrea_Town/Ba...
1911 census http://www.census.nationalarchives.ie/pages/1911/Galway/Loughrea_Urban/B...
The 1855 Griffiths Valuation head of household listing for Loughrea civil parish does not show any Graham records but there was a Thomas Byrne in the town. Not sure if he was connected to Margaret. https://www.failteromhat.com/griffiths/galway/loughrea.php
Roger McDonnell