Maximum atom-bond connectivity index for graphs with given chromatic number

Let $G$ be an $n$-vertex connected graph with chromatic number $\chi\geq 3$. For $t\geq 1$, let $T_{n,t}$ denote the complete $t$-partite graph of order $n$ whose partition sizes differ by at most $1$. Chromatic-number extremal conjecture.

$$ABC(G)\leq ABC(T_{n,\chi})$$

with equality if and only if $G\cong T_{n,\chi}$. The result is proved in the paper for chromatic number $2$; the assertion for chromatic number at least $3$ is suggested by computational results and combinatorial intuition and remains open.