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.
@article{10_37236_11183,
author = {Mingqing Zhai and Ruifang Liu and Jie Xue},
title = {A unique characterization of spectral extrema for friendship graphs},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {3},
doi = {10.37236/11183},
zbl = {1496.05107},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11183/}
}
TY - JOUR
AU - Mingqing Zhai
AU - Ruifang Liu
AU - Jie Xue
TI - A unique characterization of spectral extrema for friendship graphs
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/11183/
DO - 10.37236/11183
ID - 10_37236_11183
ER -
%0 Journal Article
%A Mingqing Zhai
%A Ruifang Liu
%A Jie Xue
%T A unique characterization of spectral extrema for friendship graphs
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/11183/
%R 10.37236/11183
%F 10_37236_11183
Mingqing Zhai; Ruifang Liu; Jie Xue. A unique characterization of spectral extrema for friendship graphs. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/11183
