Spectral radius and clique partitions of graphs

TL;DR

This paper establishes spectral bounds on clique partitions and edge-disjoint cliques in graphs, characterizing extremal graphs as block graphs of Steiner designs and regular graphs with clique decompositions.

Contribution

It introduces new spectral bounds relating graph eigenvalues to clique partition sizes and characterizes extremal graphs achieving these bounds.

Findings

Lower bounds on clique partition sizes based on spectral radius and degree

Spectral upper bounds on maximum edge-disjoint t-cliques

Extremal graphs are block graphs of Steiner 2-designs and regular graphs with K_t-decompositions

Abstract

We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint $t$-cliques. The extremal graphs attaining the bounds are exactly the block graphs of Steiner $2$-designs and the regular graphs with $K_t$-decompositions, respectively.