Spectral extrema of graphs with fixed size: Forbidden star forests

The spectral radius of a graph G , denoted by ρ(G), is the largest eigenvalue of its adjacency matrix. The Brualdi-Hoffman-Turán type problem is to determine the maximum spectral radius among all m -edge graphs which do not contain specific forbidden subgraphs. Denote by S_ℓ the star on ℓ+1 vertices. Let F be a star forest, where F=∪_i=1^kS_ℓi with k≥2 and ℓ_i≥1 for i∈[k]. In this paper, we study the Brualdi-Hoffman-Turán type problem for star forests, and prove that if G is an F -free graph with size m , then its spectral radius satisfies ρ(G)≤12(k−2+4m−k²+2k), with equality if and only if G=K_k−1∨(mk−1−k−22)K₁, provided that m≥(2k−1)²(∑i=1kℓi+k−2)².