A sharp fixed-size spectral bound for $k{K}_3$-free graphs

Abstract

For a fixed integer $k\ge 2$, we establish a sharp adjacency-spectral upper bound for sufficiently large $m$-edge $k{K}_3$-free graphs. We prove \[ λ(G)\le (k-1)+\sqrt{m-k(k-1)}. \] 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 $k\ge 3$. 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 Erdős--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.

32 pages, Comments are welcome