What is the scriptural basis for a monarchical episcopate?

Acts 1:20 speaks of what to do with the office of a heretical bishop (Judas Iscariot in this particular case):

For it is written in the book of Psalms ^[108:8]. Be their habitation made desert, and be there none to dwell in it. And his Bishopric* let another take.

^{{*A bishopric can refer to the province of a bishop (diocese) or to his office.}}

Thus, a heretical bishop is to be replaced, leaving only one bishop to occupy a bishopric at a time.

Some hint may be in Mt 16,18-19: (NKJV)

18 And I also say to you that you are Peter, and on this rock I will build My church, and the gates of Hades shall not prevail against it. 19 And I will give you the keys of the kingdom of heaven, and whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven."

I'm aware that's not a sufficient answer, but it can be a good start.

The scriptural basis on which Catholicism support the church polity of monarchial episcopate are the following scriptural passages:

Acts 21:18

On the following day Paul went in with us to James, and all the elders were present-(ESV)

In Catholicism, being an elder and being a bishop are two different offices.Futhermore, in Catholicism, St. James was believed to be a bishop. Therefore, Acts 21:18 is an explicit Bible verse that supports monarchial episcopate when understood in terms of overall Catholic teachings.

Titus 1:5

This is why I left you in Crete, so that you might put what remained into order, and appoint elders in every town as I directed you—(ESV)

Catholics believe that bishoprick and eldership aren't the same.Catholics believe that St. Titus was a bishop.In this verse, St. Titus appoints elders. In Catholicism, appointing elders is the function of a bishop.When read in Catholic lenses, Titus 1:5 is an another Bible verse that clearly supports monarchial episcopacy.

1 Timothy 5:17-22

Let the elders who rule well be considered worthy of double honor, especially those who labor in preaching and teaching. 18 For the Scripture says, “You shall not muzzle an ox when it treads out the grain,” and, “The laborer deserves his wages.” 19 Do not admit(you-singular) a charge against an elder except on the evidence of two or three witnesses. 20 As for those who persist in sin, rebuke (you-singular) them in the presence of all, so that the rest may stand in fear. 21 In the presence of God and of Christ Jesus and of the elect angels I charge you to keep these rules without prejudging, doing nothing from partiality. 22 Do not be hasty in the laying on of hands(you-singular), nor take part in the sins of others; keep yourself pure -(ESV).

St. Timothy is a bishop according to Catholicism.In this text, he possesses exclusive authority over the Ephesian elders.1 Timothy 5:17-22 supports monarchial episcopacy when read in light of Catholic theology.

Conclusion

Ample scriptural passages provided cogent biblical evidence for monarchial episcopate. This is only true when read in light of Catholic interpretation of eldership and bishoprick, though.

References:

http://www.biblicalcatholic.com/apologetics/a80.htm

http://www.newadvent.org/cathen/02581b.htm

St. Robert Bellarmine's De Romano Pontifice (On the Roman Pontiff), recently translated into English for the first time, treats the question "Should the Ecclesiastical Government be a Monarchy?" in book 1, chapters 5 through 9:

5. The Second Question is Proposed; Should the Ecclesiastical Government be a Monarchy?
6. That the Government of the Church Should not be a Democracy
7. That Ecclesiastical Government Should not be in the Power of Secular Princes
8. That Ecclesiastical Government Should not be Chiefly in the Power of Bishops
9. Why the Ecclesiastical Government Should Particularly be a Monarchy

St. Robert quotes Holy Scriptures much more copiously in support of these theses than can be summarized here; but—besides his Old Testament examples of monarchies, of Abraham, Moses, Aaron, et al. being monarchs, and the leaders of the twelve tribes being a prefiguration of the twelve Apostles—probably the single best Scriptural evidence of a monarchical constitution of the Church that St. Robert gives in ch. 9 is this part of John 10:16:

John 10:16 …There will be one flock and one shepherd.

To which St. Robert commentates:

Moreover, one sheepfold also requires one shepherd, as is gathered from the Gospel: “There will be one flock and one shepherd.” [John X.] It must be noted in passing, that “one shepherd” can be understood concerning a secondary pastor, namely Peter and his successors, as Cyprian expresses it. For when the Lord said he has other flocks and other sheep who are not of this fold, he speaks on the Gentile people and the people of the Jews: but he teaches that he has among the nations many elect, who either are already faithful, or certainly are going to be, and nevertheless they do not pertain to that Judaic people.

If it is a question of the shepherd of God, the people of the Jews and gentiles were always one flock, and one God was their shepherd: nevertheless there was not always one flock and one shepherd with respect to the governance of the human race; nor indeed the gentiles, or those among them pertaining to the Church, ruled by the priest of the Jews. But Christ wished after his arrival, that one flock be made from each people, and all men to be governed by one shepherd.

and

The similitude of the house and the boat remain, and indeed every house has one Lord and one steward, according to that of Luke’s gospel: “Who do you think is the faithful dispenser, and prudent, whom the Lord constituted over his household?” [Luke XII.] These words are said for Peter, and about Peter himself, since a little before the Lord had said to him: “Blessed are those servants, whom the Lord will discover watching when he will have come.” Peter asked: “O Lord, you speak to us this parable, can it be for all? The Lord responded to Peter: ‘Who do you think is the faithful and prudent dispenser? Whom the Lord constituted over his house?” It is just as if he were to say, where O Peter I say in the first place, it behooves you therefore to consider, what is required in a faithful and prudent steward, whom the Lord will establish over his household.

Chapters 1 through 4 are devoted to the question "What Might be the Best System of Government?" in which he shows that monarchy is simply the best, but "That Monarchy Mixed with Aristocracy and Democracy, Should be More Useful in this Life." He frequently quotes and refutes Calvin, who argued that aristocracy is better than monarchy even for ecclesiastical governance.

St. Robert's argument in a nutshell is:

1. Monarchy is the best.
2. God would not have chosen an inferior form of government for His Church.
3. Therefore, the Church is monarchical.

The Scripture from which he quoted or referenced in ch. 9: 1 Peter II. Matt. XVII. John I. Hebrews VIII. Job LX; Isaiah XIV. Ezechial XXVIII: 13. 1 Corinth. X. Song of Songs, VI. Ibid, VII. Daniel II; John I, 1 Timoth. III; 1 Peter III. Apoc. XXI; 2 Cor. XI; Ephes. V; often in the Song of Songs. Song of Songs VII. Ibid, V. John X. Luke XII. 1 Timothy 5. Romans X. Ephesians IV. 2 Cor. XI. Psalm XLIV. 1 Corinthians III. Psalm LXXXI. 2 Cor. XI. Daniel II: 37-38. Isaiah XLV:1. Luke II:1. 1 Cor. 3.