Maxima of the Q-index: graphs with bounded clique number

TL;DR

This paper establishes a precise upper bound on the spectral radius of the signless Laplacian for graphs with a given order and clique number, confirming a conjecture and characterizing extremal graphs.

Contribution

It provides a tight upper bound on the signless Laplacian spectral radius for graphs with bounded clique number, confirming a conjecture and identifying extremal structures.

Findings

The maximum eigenvalue q(G) is at most 2(1-1/r)n for graphs with clique number r.

Equality is achieved by complete r-partite regular graphs.

The result confirms a conjecture of Hansen and Lucas.

Abstract

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number r, then the largest eigenvalue q(G) of the matrix Q=A+D satisfies q(G)<= 2(1-1/r)n. If G is a complete regular r-partite graph, then equality holds in the above inequality. This result confirms a conjecture of Hansen and Lucas.