Braixen

Summary
This paper tackles spectral extremal problems for planar and outerplanar graphs excluding the “bicyclic” graph $C_{k,l}$ — two cycles sharing exactly one vertex, with $l \ge k \ge 3$. Building on Tait–Tobin’s structural framework and Yin–Li’s prior work on $B_{t,l}$-free graphs, it determines the corresponding equation in the paper and the corresponding equation in the paper for all such $k,l$ and sufficiently large $n$, identifying unique extremal graphs in each case: $K_2 + H_{\mathcal{P}}(\cdot,\cdot)$ for planar graphs (with three distinct cases depending on $l$ relative to $k$), and $K_1 + H_{\mathcal{OP}}(l-2,l-2)$ for outerplanar graphs.

Mathematical/empirical assessment
The proof strategy is sound: leveraging known lemmas (e.g., lm1, lm8) to force extremal graphs to contain $K_{2,n-2}$ or $K_{1,n-1}$, then analyzing the induced subgraph on the remaining vertices via path decompositions and $(s_1,s_2)$-transformations (lm3, lm10). However, I am not fully convinced by the justification for the threshold the corresponding equation in the paper in Theorem 1. The derivation of that cubic lower bound (e.g., in Claim 5’s inequality chain) relies on bounding $\rho(G)$ below by $\sqrt{2n-4}$, but this estimate is too crude for the required precision — the Rayleigh quotient argument uses the corresponding equation in the paper and the corresponding equation in the paper coefficients from lm2, yet no verification is given that these constants hold uniformly across all claimed $n$-ranges, especially when $k$ grows. Similarly, the outerplanar threshold in Theorem 2 invokes the corresponding equation in the paper, but the numerator’s dependence on both $k$ and $l$ isn’t justified by any stability or perturbation analysis — it appears pulled from algebraic rearrangement without empirical or asymptotic validation.

Strengths
The paper cleanly extends the spectral extremal paradigm to asymmetric bicyclic forbidden subgraphs — a natural and previously unaddressed generalization beyond $C_{l,l}$ or $B_{t,l}$. Its case-splitting in Theorem 1 ($l \le 2k-2$, $l \in \{2k-1,2k\}$, $l \ge 2k+1$) reflects genuine structural differences in how $C_{k,l}$ embeds into $K_2 + H$, and the use of $H_{\mathcal{P}}$ and $H_{\mathcal{OP}}$ definitions ties tightly to prior work. The outerplanar result is notably cleaner and more uniform than the planar one — a strength, not a weakness — and its proof avoids the most delicate combinatorial casework.

Concerns
I am not fully convinced the extremal uniqueness claims hold for all $l \ge k \ge 3$ as stated. For instance, when $k=3$, $l=7$, the condition $l \ge 2k+1$ applies, so Theorem 1 prescribes the corresponding equation in the paper. But $H_{\mathcal{P}}(2,2)$ consists of disjoint $P_2$’s — i.e., edges — and $K_2 + (\text{many } P_2)$ contains many $C_{3,7}$ configurations (e.g., using one $P_2$ for the triangle and a longer induced path across others). The argument in Claim 7 assumes the corresponding equation in the paper, but doesn’t rule out alternative path-length distributions (e.g., one long path plus many short ones) that might evade $C_{k,l}$ while yielding higher spectral radius — especially since the transformation lemmas (lm3, lm10) require $n$ exponentially large in $k$ or $l$, yet the theorem only assumes polynomial lower bounds.

Final decision
Weak accept