Spectral extrema of graphs with bounded clique number and matching number

For a set of graphs F, let ex(n,F) and spex(n,F) denote the maximum number of edges and the maximum spectral radius of an n -vertex F-free graph, respectively. Nikiforov (LAA, 2007) gave the spectral version of the Turan Theorem by showing that spex(n, Kk+1) = lambda(Tk(n)), where Tk(n) is the k -partite Turan graph on n vertices. In the same year, Feng, Yu and Zhang (LAA) determined the exact value of spex(n,Ms+1), where Ms+1 is a matching with s + 1 edges. Recently, Alon and Frankl (arXiv :2210 .15076) gave the exact value of ex(n, {Kk+1, Ms+1}). In this article, we give the spectral version of the result of Alon and Frankl by determining the exact value of spex(n, {Kk+1, Ms+1}) when n is large. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

In this article, the author determines the exact value of spex(n, {Kk+1, Ms+1}) for large n, which provides the spectral version of the result by Alon and Frankl.