Signless Laplacian spectral radius of graphs without short cycles or long cycles.

The signless Laplacian spectral radius of a graph G , denoted by q (G) , is the largest eigenvalue of its signless Laplacian matrix. In this paper, we investigate extremal signless Laplacian spectral radius for graphs without short cycles or long cycles. Let G (m , g) be the family of graphs on m edges with girth g and H (m , c) be the family of graphs on m edges with circumference c. More precisely, we obtain the unique extremal graph with maximal q (G) in G (m , g) and H (m , c) , respectively. [ABSTRACT FROM AUTHOR]