On a conjecture related to the smallest signless Laplacian eigenvalue of graphs

Abstract

For a simple graph G, the signless Laplacian matrix of G, denoted by Q(G), is defined as D(G) + A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. By the smallest signless Laplacian eigenvalue of G, denoted by q′(G), we mean the smallest eigenvalue of Q(G). Let T(n, t) be the Turán graph on n vertices and t parts. In De Lima et al. (The clique number and the smallest Q-eigenvalue of graphs. Discrete Math. 2016;339:1744–1752), the authors posed the following conjecture:Conjecture. Let t ≥ 3 and let n be sufficiently large. If G is a K_t+1-free graph of order n and G ≠ T(n, t), then q′(G) < q′(T(n, t)).In this paper, first we disprove the above conjecture and then pose a new conjecture.

Keywords:

• Smallest signless Laplacian eigenvalue of graphs
• complete multipartite graphs

AMS Classifications:

• 05C31
• 05C50
• 15A18

Acknowledgments

The author is grateful to the referees for their helpful comments. This research was in part supported by Iran National Science Foundation (INSF) under the contract No. 98001945.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was in part supported by Iran National Science Foundation (INSF) under the contract No. 98001945.