Brualdi-Hoffman-Turán problem of the gem

Abstract

A graph is said to be $F$-free if it does not contain $F$ as a subgraph. Brualdi-Hoffman-Turán problem seeks to determine the maximum spectral radius of an $F$-free graph with given size. The gem consists of a path on $4$ vertices, along with an additional vertex that is adjacent to every vertex of the path. Concerning Brualdi-Hoffman-Turán problem of the gem, when the size is odd, Zhang and Wang [Discrete Math. 347 (2024) 114171] and Yu, Li and Peng [arXiv:2404. 03423] solved it. In this paper, we completely solve the Brualdi-Hoffman-Turán problem type problem of the gem.