In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, where k ≥ 3, reaches its upper bound 2∆(G), where ∆(G) is the largest degree of G, if and only if G is regular. Thus the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and oddbipartite. We show that an s-cycle G, as a k-uniform hypergraph...

We prove that the minimum value of the least eigenvalue of the signless Laplacian of a connected nonbipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices. © 2008 Elsevier Inc. All rights reserved. AMS classification: 05C50

In this paper we observe that the minimal signless Laplacian spectral radius is obtained uniquely at the kite graph PKn−ω,ω among all connected graphs with n vertices and clique number ω. In addition, we show that the spectral radius μ of PKm,ω (m ≥ 1) satisfies

LetB(n, r) be the set of all bicyclic graphs with n vertices and r cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in B(n, r).

In this paper, some new properties are presented to the extremal graphs with largest (signless Laplacian) spectral radii in the set of all the connected graphs with prescribed degree sequences, via which we determine all the extremal tricyclic graphs in the class of connected tricyclic graphs with prescribed degree sequences, and we also prove some majorization theorems of tricyclic graphs with...

In this paper, we determine the maximum signless Laplacian spectral radius of all graphs which do not contain small books as a subgraph and characterize extremal graphs. addition, give an upper bound intersecting quadrangles subgraph.

A cactus is a connected graph in which any two cycles have at most one vertex in common. We determine the unique graphs with maximum signless Laplacian spectral radius in the class of cacti with given number of cycles (cut edges, respectively) as well as in the class of cacti with perfect matchings and given number of cycles.

In this paper, we determine the unique graph whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with maximum degree Δ and among all non-bipartite connected graphs of order n with maximum degree Δ, respectively.