An extension of spectral Mantel's theorem on wheels

Abstract

A graph is considered wheel-free if the neighborhood of any vertex is acyclic. The extremal problems associated with wheel-free graphs have a long-standing history of research. In 1983, Gallai and Zelinka independently posed the question of determining the maximum number of triangles in an n-vertex wheel-free graph. Moving forward to 2021, Zhao, Huang and Lin investigated the maximum spectral radius of graphs within the same family of wheel-free graphs.

In this paper, we focus on graphs of fixed size that do not contain isolated vertices. In 1970, Nosal established that every graph G with m edges and a spectral radius $\rho \left(G\right)>\sqrt{m}$ contains at least one triangle. This result is known as the spectral Mantel's theorem. Nikiforov further refined this theorem by showing that any graph G with m edges and a spectral radius $\rho \left(G\right)\ge \sqrt{m}$ contains a triangle, except when G is a complete bipartite graph.

In this work, we present an extension of spectral Mantel's theorem, which asserts that every graph G with $m\ge 25$ edges and a spectral radius $\rho \left(G\right)\ge \frac{1}{2}(1+\sqrt{4m-5})$ contains a wheel, unless G is a book (possibly missing an edge).