Spectral extremal graphs for disjoint cliques

TL;DR

This paper characterizes the graph with the maximum spectral radius among all large graphs that do not contain $k$ disjoint $(r+1)$-cliques, showing it is isomorphic to a specific join of a complete graph and a Turán graph.

Contribution

It extends classical extremal graph results by identifying the spectral extremal graph for disjoint cliques, confirming its structure for sufficiently large graphs.

Findings

Maximum spectral radius graph is isomorphic to $K_{k-1} \,\vee\, T_{n-k+1,r}$.

Unique extremal graph for large $n$ without $k$ disjoint $(r+1)$-cliques.

Generalizes previous extremal results to spectral graph theory.

Abstract

The $kK_{r+1}$ is the union of $k$ disjoint copies of $(r+1)$-clique. Moon [Canad. J. Math. 20 (1968) 95--102] and Simonovits [Theory of Graphs (Proc. colloq., Tihany, 1996)] independently showed that if $n$ is sufficiently large, then $K_{k-1}\vee T_{n-k+1,r}$ is the unique extremal graph for $kK_{r+1}$. In this paper, we consider the graph which has the maximum spectral radius among all graphs without $k$ disjoint cliques. We prove that if $G$ attains the maximum spectral radius over all $n$-vertex $kK_{r+1}$-free graphs for sufficiently large $n$, then $G$ is isomorphic to $K_{k-1}\vee T_{n-k+1,r}$.