• śrī-bādarāyaṇiḥ — Śrī Bādarāyaṇi (Śukadeva)
  Patronymic. Śukadeva is the son of Bādarāyaṇa (Vyāsadeva). The honorific is invoked at the entrance of major narrative arcs to remind the hearer of the speaker's parampara-authority — this is no folk-story; it descends from Vyāsa.
• vidarbha-adhipatiḥ — the ruler of Vidarbha
  Compound. Vidarbha is the Berar region; its kings are recurring figures across the epics (Damayantī's father Bhīma was also a Vidarbha-king). The dynastic placement matters — Rukmiṇī is born into Vedic kṣatriya pedigree of the highest rank.
• kanyā-ekā — one (sole) daughter
  The numerical 'one' is pointed — Rukmiṇī has no sisters. Gauḍīya readers note that the Bhāgavatam often gives a single bride the central frame so that her bhāva can be displayed without competition; cf. Sītā, Yaśodā as Kṛṣṇa's sole mother in Vraja.
• vara-ānanā — whose face is exceptionally beautiful
  Compound: vara (excellent) + ānanā (faced). The same vocabulary that will saturate Rukmiṇī's later letter — her face is the chapter's recurring metonym for her whole prema.

• Padaseva Pr Seminars Oriya - SB 10-52-21
  His Grace Padaseva Prabhu· Oriya

Bhīṣmaka's household is a stage set before the play begins — five sons, one daughter, a great kingdom — and the audience is told their names before the prema-drama opens. The Bhāgavatam introduces Rukmiṇī through the standard dynastic camera, then pulls in close enough to read her thoughts.