Esmé Stewart, 1st Duke of Lennox
Esmé Stewart, 1st Duke of Lennox, 1st Earl of Lennox( 154226 May 1583) was the son of John Stewart, 5th Lord of Aubigny, who was the younger brother of Matthew Stewart, 4th Earl of Lennox, and his wife Anne de la Queuille.
Sir James Melville described him as" of nature, upright, just, and gentle".
John Stewart, 3rd Earl of Lennox
John Stewart, 3rd Earl of Lennox( c. 14904 September 1526, Linlithgow, West Lothian) was a prominent Scottish magnate.
He was the son of Matthew Stewart, 2nd Earl of Lennox and Lady Elizabeth Hamilton, daughter of James Hamilton, 1st Lord Hamilton and Mary Stewart, Princess of Scotland, daughter of King James II of Scotland.
The Earl of Lennox led an army to Linlithgow with the intention of liberating the young King James V of Scotland from the pro-English Douglases.
He was defeated by a smaller force led by James Hamilton, 1st Earl of Arran, at the Battle of Linlithgow Bridge.
He survived the battle and was taken captive, only to be subsequently murdered by James Hamilton of Finnart.
Lennox was succeeded by his son, Matthew Stewart, the father of Henry Stewart, Lord Darnley, and the grandfather of King James VI of Scotland.
Margaret, Countess of Lennox
Margaret, Countess of Lennox was the daughter of Domhnall, Earl of Lennox, who died in about 1364.
She married Walter de Fasselane, a cousin, who also descended from previous Earls of Lennox.
Margaret and Walter's children were:
Robert Stewart, 1st Earl of March
Robert Stewart, 1st Earl of Lennox then 1st Earl of March( c. 152229 March 1586) was a Scottish nobleman of the family of Stewart of Darnley.
Charles Stuart, 1st Earl of Lennox
Charles Stuart, 1st (or 5th) Earl of Lennox (April or May 1557 – April 1576), was the fourth son of Matthew Stewart, 4th Earl of Lennox and Margaret Douglas, daughter of Margaret Tudor and granddaughter of King Henry VII of England.
His brother was Henry Stuart, Lord Darnley.
John Stewart, 1st Earl of Lennox
John Stewart, 1st Earl of Lennox( before 14308 Jul/ 11 Sep 1495) was known as Lord Darnley and later as the Earl of Lennox.
James Stuart, 2nd Earl of Bute
James Stuart, 2nd Earl of Bute( before 1696 – 28 January 1723) was the son of James Stuart, 1st Earl of Bute and Agnes Mackenzie.
Elizabeth Stuart, Countess of Lennox
Elizabeth Stuart, Countess of Lennox née Cavendish (31 March 1555 – 16 January 1582) was an English noblewoman and the wife of Charles Stuart, 1st Earl of Lennox.
She was the mother of Arbella Stuart, a close claimant to the English and Scottish thrones.
Matthew Stewart, 4th Earl of Lennox
Matthew Stewart, 4th Earl of Lennox (21 September 15164 September 1571), was the fourth Earl of Lennox, and a leader of the Catholic nobility in Scotland.
He was the son of John Stewart, 3rd Earl of Lennox and Lady Elizabeth Stewart, daughter of John Stewart, 1st Earl of Atholl.
His grandson was King James VI of Scotland and I of England.
Charles Stuart, 1st Baron Stuart de Rothesay
Charles Stuart, 1st Baron Stuart de Rothesay (2 January 1779 – 6 November 1845), known as Sir Charles Stuart between 1812 and 1828, was a British diplomat.
He was twice Ambassador to France and also served as Ambassador to Russia between 1841 and 1844.

Question: Who is the paternal grandfather of Charles Stuart, 1St Earl Of Lennox?
Answer: John Stewart, 3rd Earl of Lennox