Spectral extremal graphs for the bowtie

Abstract

Let Fk be the (friendship) graph obtained from k triangles by sharing a common vertex. The Fk-free graphs of order n which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that n is sufficiently large. In this paper, we get rid of the condition on n being sufficiently large if k=2. The graph F2 is also known as the bowtie. We show that the unique n-vertex F2-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if n≥7, and the condition n≥7 is tight. Our result is a spectral generalization of a theorem of Erdős, Füredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that ex(n,F2)=⌊n2/4⌋+1. Moreover, we study the spectral extremal problem for Fk-free graphs with given number of edges. In particular, we show that the unique m-edge F2-free spectral extremal graph is the join of K2 with an independent set of m−12 vertices if m≥8, and the condition m≥8 is tight.