Let Fk be the (friendship) graph obtained from k triangles by sharing a common vertex. In 2024, Li, Lu, Peng [Discrete Mathematics
346(2023)] show that the unique n-vertex F2 -free spectral extremal
graph is the balanced bipartite graph adding an edge in smaller part
if n ≥7. Following their result, we show that the unique n-vertex
F3-free spectral extremal graph is the balanced complete bipartite
graph adding two disjoint K3 in the vertex part with smaller size if
n > 360.
References
[1]
Bondy, J.A. and Murty, U.S.R. (2008) Graph Theory, Graduate Texts in Mathematics, vol. 24. Springer.
[2]
Erdos, P., Furedi, Z., Gould, R.J. and Gunderson, D.S. (1995) Extremal Graphs for Intersecting Triangles. Journal of Combinatorial Theory, Series B, 64, 89-100.https://doi.org/10.1006/jctb.1995.1026
[3]
Cioab, S., Feng, L., Tait, M. and Zhang, X. (2020) The Maximum Spectral Radius of Graphs without Friendship Subgraphs. The Electronic Journal of Combinatorics, 27, Article No. P4.22.https://doi.org/10.37236/9179
[4]
Zhai, M., Liu, R. and Xue, J. (2022) A Unique Characterization of Spectral Extrema for Friendship Graphs. The Electronic Journal of Combinatorics, 29, Article No. P3.32.https://doi.org/10.37236/11183
[5]
Li, Y., Lu, L. and Peng, Y. (2023) Spectral Extremal Graphs for the Bowtie. Discrete Mathematics, 346, Article ID: 113680. https://doi.org/10.1016/j.disc.2023.113680
[6]
Chen, Y., Li, S., Zhang, L. and Zhang, M. (2025) Spectral Extrema of F2-Free Graphs with Given Size Revisited. Discrete Applied Mathematics, 375, 294-307.https://doi.org/10.1016/j.dam.2025.06.007
[7]
Yu, L., Li, Y. and Peng, Y. (2025) Spectral Extremal Graphs for Fan Graphs. Discrete Mathematics, 348, Article ID: 114391.https://doi.org/10.1016/j.disc.2024.114391
[8]
Gao, J. and Li, X. (2026) Spectral Radius of Graphs of Given Size with a Forbidden Fan Graph F6. Discrete Mathematics, 349, Article ID: 114695. https://doi.org/10.1016/j.disc.2025.114695
[9]
Li, S., Zhao, S. and Zou, L. (2025) Spectral Extrema of Graphs with Fixed Size: Forbidden a Fan Graph, a Friendship Graph, or a Theta Graph. Journal of Graph Theory, 110, 483-495. https://doi.org/10.1002/jgt.23287
[10]
Zhang, Y. and Wang, L. (2024) On the Spectral Radius of Graphs without a Gem. Discrete Mathematics, 347, Article ID: 114171. https://doi.org/10.1016/j.disc.2024.114171
[11]
Li, Y., Feng, L. and Peng, Y. (2025) A Spectral Erdos-Faudree-Rousseau Theorem. Journal of Graph Theory, 110, 408-425. https://doi.org/10.1002/jgt.23280
[12]
Hoffman, A.J. and Smith, J.H. (1975) Recent Advances in Graph Theory. Academic Praha.
[13]
Csikvri, P. (2009) On a Conjecture of V. Nikiforov. Discrete Mathematics, 309, 4522-4526. https://doi.org/10.1016/j.disc.2009.02.013
[14]
Chvtal, V. and Hanson, D. (1976) Degrees and Matchings. Journal of Combinatorial Theory, Series B, 20, 128-138.https://doi.org/10.1016/0095-8956(76)90004-6
