The minimum Graovac-Ghorbani index conjecture for the B2 family

Let $B_2(n)$ denote the family of bicyclic graphs specified in the paper, and let $G\in B_2(n)$ have order $n\geq 11$. The Graovac-Ghorbani atom-bond connectivity index is denoted by $ABC_{GG}(G)$. B2-family minimum conjecture.

$$ABC_{GG}(G)\geq ABC_{GG}\bigl(B_2(3,1,n-3)\bigr).$$

Equality holds if and only if $G\cong B_2(3,1,n-3)$. This is the proposed extremal description for the minimum index within the $B_2(n)$ family, for both odd and even $n\geq 11$.