Which multicone graphs are determined by their signless Laplacian spectrum? (The proof of a conjecture)

Abstract

A multicone graph is defined to be the join of a clique and a regular graph. Let K_m and denote a complete graph on m vertices and the complement of a complete graph on n vertices, respectively. In [X. Liu and P. Lu, Signless Laplacian spectral characterization of some joins, Electron. J. Linear Algebra 30.1 (2015)], it have been conjectured that the multicone graphs are determined by their signless Laplacian spectrum. In this short note, we prove this conjecture for n ≠ 3.

Subject Classification:

• (2010) 05C50
• 20D30

Keywords:

• DS graph
• Multicone graph
• Signless Laplacian spectrum