Sheikh Sulaymân al-Asqah
If everything is exactly as you have described it in your question, then you have done nothing wrong. You have only sought what is your right, what you could reasonably expect to be entitled to. You have taken the necessary steps to secure your rights when the circumstances you have had to cope with were unfair and full of injustice, so that only those who are well-connected can secure entry into the university's graduate program.
What you describe as "connections" fall under the broader category of what is referred to in Islamic Law as intercession. According to Islamic Law, seeking intercession is allowed in securing one's legitimate rights or in warding off injustice. Likewise, it is a good deed for the person who intercedes o someone's behalf with the sincere intention of championing that person's rights and fighting against injustice.
Likewise, intercession is unlawful and sinful when it is used to secure something that someone is not entitled to, or it is used to misappropriate the rights of others.
Allah says: "Whoso intercedes in a good cause will have the reward thereof, and whoso intercedes in an evil cause will bear the consequence thereof. Allah oversees all things. " [Sûrah al-Nisâ': 85]
The Prophet (peace be upon him) said: "Intercede and receive blessings." [Sahîh al-Bukhârî (1432) and Sahîh Muslim (2627)]
And Allah knows best.
Source: Islam Today