Spectral extremal graphs for edge blow-up of star forests

TL;DR

This paper characterizes the extremal graphs with maximum spectral radius that avoid containing edge blow-ups of star forests, extending understanding of spectral extremal problems for complex graph structures.

Contribution

It identifies the extremal graphs with maximum spectral radius for avoiding edge blow-ups of star forests when the number of vertices is large.

Findings

Maximum spectral radius graphs are extremal for edge blow-up of star forests.

Established the structure of extremal graphs for large n.

Extended spectral extremal graph theory to complex graph blow-ups.

Abstract

The edge blow-up of a graph $G$, denoted by $G^{p+1}$, is obtained by replacing each edge of $G$ with a clique of order $p+1$, where the new vertices of the cliques are all distinct. Yuan [J. Comb. Theory, Ser. B, 152 (2022) 379-398] determined the range of the Tur\'{a}n numbers for edge blow-up of all bipartite graphs and the exact Tur\'{a}n numbers for edge blow-up of all non-bipartite graphs. In this paper we prove that the graphs with the maximum spectral radius in an $n$-vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when $n$ is sufficiently large.