The planar Turan number of double star S_(3,5)

Abstract: Given a graph H and a positive integer n, the planar Turan number of H, denoted by exp(n, H), is the maximum number of edges in an n-vertex H-free planar graph.D.Ghosh, et al.initiated the topic of double stars S_(k,l). Recently Xu et al.[AIMS Mathematics, 2025, 10(1): 1628-1644.] mentioned that exp(n, S_(3,5)) is still unknown.In this paper, we first establish that the planar Turan number S_(3,5) satisfies exp(n, S_(3,5)) <= 23n/8 - 9/2 for all n >= 2. The upper bound is tight for n = 12.