The essence of Hopf Algebra is a weird ‘Yin-Yang‘ view of every algebra concept.
the ‘co-‘ notion represents the ‘Yin‘ opposite of the ‘Yang‘ notion.
[Analogy: Set & coSet 倍集， l’ensemble à gauche ou à droite]
Hopf Algebra is applied in Quantum Group for Quantum Physics, Yang-Baxtor equation, Combinatorics and Topology, etc.
If you find a structure which has a combined properties of Group, Ring and Vector Space, then it could be a Hopf Algebra.
Applied it in Graphs, we can re-discover the Euler’s (E-V+F = 2 ) formula, Möbius function, and Symmetric Group (Sn).
Note: Excellent 30-lecture series by Prof Federico Ardila (San Francisco University) on Hopf Algebra: