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

Abstract

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.