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Given a graph F, let SPEXP (n, F) be the set of graphs with the maximum spectral radius among all F-free n-vertex planner graph. In 2017, Tait and Tobin proved that for sufficiently n, K₂+P₍-₂ is the unique graph with the maximum spectral radius over all n-vertex planner graphs. In this paper, focusing on SPEXP (n, K₂+H) in which H is a linear forest, we prove that SPEXP (n, K₂+H) =\2K₁+C₍-₂\ when H \pK₂, P₃, Iq\ (p1, q 3), where Kₙ, Pₙ, Iₙ are complete graph, path and empty graph of order n, respectively. When H contains a P₄, we prove that 2K₁+C₍-₂ SPEXP (n, K₂+H) and also provide a structural characterization of graphs in SPEXP (n, K₂+H).
Xu et al. (Sun,) studied this question.