Use of the terms "Letters of Administion" and "relinquishing rights to administer" indicates no will was made or found. In these cases the court appoints an administer who is bonded. Normally, the bond is set at twice the value of the estate. The term "relinquishing rights to administer" indicates that the person originally appointed by the court was discharged. In such cases, another person is appointed.
The names of the children, or childen of any deceased children, are as recorded by the clerk of the probate court from affidavits made by the claiments. Normally, no relationships are shown and there may be nothing to show relationships. You will have to check the probate records.