for the symmetric Y-Channel
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In the symmetric Y-channel, $h_1=h_2=h_3=1$. In this case, we can rewrite the bounds we have in a simpler form. Starting from Corollary \[CSG\], we can show that the following bound holds $$\begin{aligned}
C_g\leq\overline{C}_{cs}&\triangleq3\min\{C(P),C(P_r)\}.\end{aligned}$$ Moreover, for the symmetric Y-channel we have the following upper bounds from Lemmas \[FromRelay\] and \[GUB\] respectively $$\begin{aligned}
C_g&\leq\overline{C}_{s}\triangleq2C(2P_r)\\
C_g&\leq\overline{C}_{g}\triangleq2C(4P).\end{aligned}$$ The following lower bounds are achievable in the symmetric Y-channel (Theorems \[LowerBound:DF\], \[LowerBound:FDF\] and \[LowerBound:FDF\_2User\]) $$\begin{aligned}
C_g\geq\underline{C}^{i}&=\min\left\{C(3P),\frac{3}{2}C(P_r)\right\}\\
C_g\geq\underline{C}^{ii}&=2\min\left\{\left[C\left(\frac{3P}{2}-\frac{1}{2}\right)\right]^+,C(P_r)\right\}\\
C_g\geq\underline{C}^{iii}&=2\min\left\{\left[C\left(P-\frac{1}{2}\right)\right]^+,C(P_r)\right\},\end{aligned}$$ where we used small letters in the superscript to distinguish these achievable sum rates from their counterparts in the asymmetric Y-channel. Now that we have upper and lower bounds for the sum-capacity of the symmetric Y-channel, we can upper bound the gap between them, which we denote by $\Delta_g$ $$\begin{aligned}
\label{Dg}
\Delta_g=\min\{\overline{C}_{cs},\overline{C}_s,\overline{C}_g\}-\max\{\underline{C}^i,\underline{C}^{ii},\underline{C}^{iii}\}.\end{aligned}$$ To simplify the calculation, we assume that $P=P_r$ and bound the gap for this case. The gap for arbitrary $P$ and $P_r$ is calculated numerically and plotted in Figure \[Gap3Dg\].

\[t\][$P_r$(dB)]{} \[b\][$P$(dB)]{} \[b\][$\Delta_g$]{} ![The gap $\Delta_g$ between the upper bound and lower bound for the general symmetric Y-channel. It can be seen that the gap is always less than 1.5 bits.[]{data-label="Gap3Dg"}](gap_general.eps "fig:"){width="\columnwidth"}

In the symmetric Y-channel with $P=P_r$, then we can 