Let $B_{n,k}$ be the graph obtained from the balanced complete bipartite graph on $n$ vertices by deleting a matching of size $k$. If $G$ is an $n$-vertex triangle-free graph with $\kappa(\comp G)\geq k$, we prove that $\rhoA(G)\leq\rhoA(B_{n,k})$ for $n\geq4k+2$, with equality precisely when $G\cong B_{n,k}$, and we compute $\rhoA(B_{n,k})$ explicitly. We also solve the bipartite problem for every $n\geq2k+1$, determine the boundary value $\operatorname{spex}_{\kappa}(2k,K_3;k)=k-1$, and settle the full problem for $k=2$. In particular, $B_{n,2}$ is uniquely extremal exactly from order $6$ onward. For $k=1$, equivalently when the complement is connected, $B_{n,1}=K_{\ceil{n/2},\floor{n/2}}-e$ is uniquely extremal for every $n\geq3$.
For a fixed integer $k\ge2$, we establish a sharp adjacency-spectral upper bound for sufficiently large $m$-edge $kK_3$-free graphs. We prove \[ \lambda(G)\le (k-1)+\sqrt{m-k(k-1)}. \] Moreover, equality holds precisely when $(2k-1)\mid m$ and, up to isolated vertices, $G$ is the join of $K_{2k-1}$ with an independent set of $m/(2k-1)-(k-1)$ vertices. The case $k=2$ was previously known; our argument establishes every fixed $k\ge3$. The proof requires information beyond first-order spectral stability. We derive an exact nonnegative defect identity at a maximum Perron vertex, use it to bound the entire outer layer by a constant, and reduce the remaining graph to a bounded core with finitely many independent twin classes. A Perron-vector concentration identity and the Erd\H{o}s--Gallai matching theorem then force the unique extremal core. A nearly extremal family lies only $\Theta(m^{-1/2})$ below the target, showing why an exact second-order analysis is necessary.
Joyentanuj Das, V. Yamini· 1 citation
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