2025/09/23 by Ruiling Zheng, Zheng, Ruiling, G. Zhang +1
Mathematics · Computer Science · #Graph theory and applications #Limits and Structures in Graph Theory #Advanced Graph Theory Research
paper · pdf · doi:10.48550/arxiv.2509.18594
A graph is sad to be H-free if it does not contain H as a subgraph. Let H(k,3) be the graph formed by taking a cycle of length k and a triangle on a common vertex. Li, Lu and Peng [Discrete Math. 346 (2023) 113680] proved that if G is an H(3,3)-free graph of size m ≥ 8, then the spectral radius ρ(G) ≤ (1+√(4 m-3))/(2) with equality if and only if G ≅ S(m+3)/(2), 2, where S(m+3)/(2), 2=K2 \vee (m-1)/(2)K1. Note that the bound is attainable only when m is odd. Recently, Pirzada and Rehman [Comput. Appl. Math. 44 (2025) 295] proved that if G is an \H(3,3),H(4,3)\-free graph of even size m ≥ 10, then ρ(G) ≤ ρ′(m) with equality if and only if G ≅ S(m+4)/(2), 2-, where ρ′(m) is the largest root of x4-m x2-(m-2) x+(m)/(2)-1=0, and S(m+4)/(2), 2- is the graph obtained from S(m+4)/(2), 2 by deleting an edge incident to a vertex of degree two. In this paper, we improve the result of Pirzada and Rehman by showing that if G is an H(4,3)-free graph of even size m ≥ 38 without isolated vertices, then ρ(G) ≤ ρ′(m) with equality if and only if G ≅ S(m+4)/(2), 2-.