Spectral extremal graphs for intersecting cliques - Desai, Dheer Noal; Kang, Liying; Li, Yongtao; Ni, Zhenyu; Tait, Michael; Wang, Jing (Linear algebra and its applications, 2022)
The (k,r)-fan is the graph consisting of k copies of the complete graph Kr which intersect in a single vertex, and is denoted by Fk,r. Erdős et al. (1995) [14] determined the maximum number of edges in an n-vertex graph that does not contain Fk,3 as a subgraph. Furthermore, Chen et al. (2003) [5] proved the analogous result on Fk,r for the general case r≥3. In this paper, we show that for sufficiently large n, the graphs of order n that contain no copy of Fk,r and attain the maximum spectral radius are also edge-extremal. That is, such graphs must have ex(n,Fk,r) edges.