Maximum clique spectral radii of F2,3-free graphs and {F2,4,K1∨R(K3),K5}-free graphs

For integers 𝑘 ≥ 1 and 𝑟 ≥ 3 , let 𝐹 𝑘 , 𝑟 denote the graph consisting of 𝑘 copies of 𝐾 𝑟 sharing exactly one common vertex. In this paper, we study two clique-spectral extremal problems associated with intersecting cliques. We prove that the maximum 3-clique spectral radius of an 𝑛 -vertex 𝐹 2 , 3 -free graph is attained uniquely by 𝐾 2 ∨ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ 𝐾 𝑛 − 2 for 𝑛 ≥ 8 . We also determine the maximum 4-clique spectral radius among 𝑛 -vertex { 𝐹 2 , 4 , 𝐾 1 ∨ 𝑅 ⁡ ( 𝐾 3 ) , 𝐾 5 } -free graphs for 𝑛 ≥ 6 , and show that it is attained uniquely by 𝐾 2 ∨ 𝑇 2 ⁢ ( 𝑛 − 2 ) , where 𝑅 ⁡ ( 𝐾 3 ) denotes the graph obtained from 𝐾 3 by replacing each edge with a triangle.