On a conjecture of spectral extremal problems

Abstract

For a simple graph F, let Ex(n,F) and Exsp(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 Turán graph Tn,r is the complete r-partite graph on n vertices where its part sizes are as equal as possible. Cioabă, 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 Ex(n,F) are Turán graphs plus O(1) edges. Then Exsp(n,F)⊂Ex(n,F) for sufficiently large n. In this paper we consider the graph F such that the graphs in Ex(n,F) are obtained from Tn,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 Ex(n,F) for n large enough. Then Cioabă, Desai and Tait's conjecture is completely solved. Furthermore, we give a stronger result.