On a conjecture of spectral extremal problems

Abstract: For a simple graph $F$, let $\mathrm{Ex}(n, F)$ and $\mathrm{Ex_{sp}}(n,F)$ denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph $F$, respectively. The Tur\'an graph $T_{n,r}$ is the complete $r$-partite graph on $n$ vertices where its part sizes are as equal as possible. Cioab\u{a}, Desai and Tait [The spectral radius of graphs with no odd wheels, European J. Combin., 99 (2022) 103420] posed the following conjecture: Let $F$ be any graph such that the graphs in $\mathrm{Ex}(n,F)$ are Tur\'{a}n graphs plus $O(1)$ edges. Then $\mathrm{Ex_{sp}}(n,F)\subset \mathrm{Ex}(n,F)$ for sufficiently large $n$. In this paper we consider the graph $F$ such that the graphs in $\mathrm{Ex}(n, F)$ are obtained from $T_{n,r}$ by adding $O(1)$ edges, and prove that if $G$ has the maximum spectral radius among all $n$-vertex graphs not containing $F$, then $G$ is a member of $\mathrm{Ex}(n, F)$ for $n$ large enough. Then Cioab\u{a}, Desai and Tait's conjecture is completely solved.