A Unique Characterization of Spectral Extrema for Friendship Graphs

  • Mingqing Zhai
  • Ruifang Liu
  • Jie Xue


Turán-type problem is one of central problems in extremal graph theory. Erdős et al. [J. Combin. Theory Ser. B 64 (1995) 89-100] obtained the exact Turán number of the friendship graph $F_k$ for $n\geq 50k^2$, and characterized all its extremal graphs. Cioabă et al. [Electron. J. Combin. 27 (2020) Paper 22] initially introduced Triangle Removal Lemma into a spectral Turán-type problem, then showed that $SPEX(n, F_k)\subseteq EX(n, F_k)$ for $n$ large enough, where $EX(n, F_k)$ and $SPEX(n, F_k)$ are the families of $n$-vertex $F_k$-free graphs with maximum size and maximum spectral radius, respectively. In this paper, the family $SPEX(n, F_k)$ is uniquely determined for sufficiently large $n$. Our key approach is to find various alternating cycles or closed trails in nearly regular graphs. Some typical spectral techniques are also used. This presents a probable way to characterize the uniqueness of extremal graphs for some of other spectral extremal problems. In the end, we mention several related conjectures.

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