Price-of-symmetrisation conjecture for average transmission

Let $G$ be a strongly connected graph in ${\mathcal{G}_n}$. For $2\leq n\leq 10$, define ${\mathcal{C}_n}$ to be the directed cycle on $n$ vertices. For $3\leq k\leq n-1$, let ${\mathcal{H}_n(k)}$ be the bag obtained from the tournament ${\mathcal{B}_k}$ by duplicating the arrow $(v_k,v_1)$ and replacing the duplicate by a path of length $n-k+1$. Let

$$r=n(\sqrt{2}-1)+8-\frac{11\sqrt{2}}{2}.$$

Price-of-symmetrisation conjecture. For $2\leq n\leq 10$,

$${\sf P}^{-}_{\sigma}(G)\leq {\sf P}^{-}_{\sigma}({\mathcal{C}_n}),$$

with equality if and only if $G\simeq {\mathcal{C}_n}$. For $n\geq 11$, set

$$k^*=\max\left({\sf P}^{-}_{\sigma}({\mathcal{H}_n(\lfloor r\rfloor)}),{\sf P}^{-}_{\sigma}({\mathcal{H}_n(\lceil r\rceil)})\right).$$

Then

$${\sf P}^{-}_{\sigma}(G)\leq {\sf P}^{-}_{\sigma}({\mathcal{H}_n(k^*)}),$$

with equality if and only if $G\simeq {\mathcal{H}_n(k^*)}$. The conjecture identifies the extremal strongly connected graphs for the price of symmetrisation with respect to the invariant $\sigma$, with cycles for orders at most $10$ and bags for larger orders; the proposed bag parameter is determined by the two integers nearest to $r$.