Spectral extremal graphs for intersecting cliques

TL;DR

This paper proves that for large graphs avoiding a specific intersecting clique structure, the graphs with the maximum spectral radius also have the maximum number of edges, linking spectral and extremal graph properties.

Contribution

It establishes that for sufficiently large graphs avoiding the $(k,r)$-fan, spectral extremality coincides with edge extremality, extending prior edge-based results to spectral properties.

Findings

Spectral extremal graphs match edge-extremal graphs for large n.

Maximum spectral radius graphs are also extremal in the number of edges.

Results generalize previous specific cases to broader $(k,r)$-fan structures.

Abstract

The $(k,r)$-fan is the graph consisting of $k$ copies of the complete graph $K_r$ which intersect in a single vertex, and is denoted by $F_{k,r}$. Erd\H{o}s, F\"uredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995) 89--100] determined the maximum number of edges in an $n$-vertex graph that does not contain $F_{k,3}$ as a subgraph. Furthermore, Chen, Gould, Pfender and Wei [J. Combin. Theory Ser. B 89 (2003) 159--171] proved the analogous result on $F_{k,r}$ for the general case $r\ge 3$.In this paper, we show that for sufficiently large $n$, the graphs of order $n$ that contain no copy of $F_{k,r}$ and attain the maximum spectral radius are also edge-extremal. That is, such graphs must have $\mathrm{ex}(n, F_{k,r})$ edges.