### In algebraic graph theory, Babai's problem was proposed in 1979 by László Babai.

Babai's problem

Let G be a finite group, let $$\operatorname {Irr} (G)}$$ be the set of all irreducible characters of G, let $$\Gamma =\operatorname {Cay} (G,S)}$$ be the Cayley graph (or directed Cayley graph) corresponding to a generating subset S of $$G\setminus \{1\}}$$, and let $$\nu$$ be a positive integer. Is the set

$$M_{\nu }^{S}=\left\{\sum _{s\in S}\chi (s)\;|\;\chi \in \operatorname {Irr} (G),\;\chi (1)=\nu \right\}}$$

an invariant of the graph $$\Gamma$$ ? In other words, does $$\operatorname {Cay} (G,S)\cong \operatorname {Cay} (G,S')}$$ imply that $$M_{\nu }^{S}=M_{\nu }^{S'}}$$ ?

BI-group (Babai Invariant group)

A finite group G is called a BI-group (Babai Invariant group) if $$\operatorname {Cay} (G,S)\cong \operatorname {Cay} (G,T)}$$ for some inverse closed subsets S and T of $$G\setminus \{1\}}$$, then M$$M_{\nu }^{S}=M_{\nu }^{T}}$$ for all positive integers $$\nu$$ .
Open problem

Which finite groups are BI-groups?

List of unsolved problems in mathematics
List of problems solved since 1995

References

Babai, László (October 1979), "Spectra of Cayley graphs", Journal of Combinatorial Theory, Series B, 27 (2): 180–189, doi:10.1016/0095-8956(79)90079-0
Abdollahi, Alireza; Zallaghi, Maysam (10 February 2019). "Non-Abelian finite groups whose character sums are invariant but are not Cayley isomorphism". Journal of Algebra and Its Applications. 18 (01): 1950013. arXiv:1710.04446. doi:10.1142/S0219498819500130.
Abdollahi, Alireza; Zallaghi, Maysam (24 August 2015). "Character Sums for Cayley Graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398.