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}.
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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}.