A Unique Characterization of Spectral Extrema for Friendship Graphs - Zhai, Mingqing; Liu, Ruifang; Xue, Jie (The Electronic Journal of Combinatorics, 2022)
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≥ 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)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.