Among gem-free graphs with m edges, m odd and m at least 23, excluding the standard extremal graph, the spectral radius is maximized only by the graph S^2_{(m+5)/2,2}.
Brualdi-Hoffman-Tur\'{a}n problem of the gem
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abstract
A graph is said to be $F$-free if it does not contain $F$ as a subgraph. Brualdi-Hoffman-Tur\'{a}n 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-Tur\'{a}n 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-Tur\'{a}n problem type problem of the gem.
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Extension on spectral extrema of gem-free graph with given size
Among gem-free graphs with m edges, m odd and m at least 23, excluding the standard extremal graph, the spectral radius is maximized only by the graph S^2_{(m+5)/2,2}.