Spectral Conditions for a Graph to be Hamilton-Connected

Hamiltonian Paths ◽

Signless Laplacian ◽

Hamilton Connected

Some spectral conditions for a graph to be Hamilton-connected in terms of the spectral radius of the adjacency matrix or signless Laplacian of the graph or its complement are established, and then the condition on the signless Laplacian spectral radius of a graph for the existence of Hamiltonian paths or cycles is given.

TWO SHARP UPPER BOUNDS FOR THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001152 ◽

2011 ◽

Vol 03 (02) ◽

pp. 185-191 ◽

Cited By ~ 6

Author(s):

YA-HONG CHEN ◽

RONG-YING PAN ◽

XIAO-DONG ZHANG

Keyword(s):

Spectral Radius ◽

Upper Bound ◽

Adjacency Matrix ◽

Laplacian Matrix ◽

Upper Bounds ◽

Extremal Graphs ◽

Signless Laplacian Matrix ◽

Signless Laplacian

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

Signless Laplacian Spectral Conditions for Hamiltonicity of Graphs

Journal of Applied Mathematics ◽

10.1155/2014/282053 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Guidong Yu ◽

Miaolin Ye ◽

Gaixiang Cai ◽

Jinde Cao

Keyword(s):

Spectral Radius ◽

Signless Laplacian ◽

Hamilton Connected

We establish some signless Laplacian spectral radius conditions for a graph to be Hamiltonian or traceable or Hamilton-connected.

On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs

Kragujevac Journal of Mathematics ◽

10.46793/kgjmat2102.299d ◽

2021 ◽

Vol 45 (02) ◽

pp. 299-307

Author(s):

HANYUAN DENG ◽

TOMÁŠ VETRÍK ◽

SELVARAJ BALACHANDRAN

Keyword(s):

Spectral Radius ◽

Adjacency Matrix ◽

Connected Graph ◽

Laplacian Matrix ◽

Harmonic Index ◽

Signless Laplacian ◽

The Matrix ◽

The Relationship

The harmonic index of a conected graph G is deﬁned as H(G) = ∑ uv∈E(G) 2 d(u)+d-(v), where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G) + A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have ( 2 || ----n----- ||{ 2 (n − 1), if n ≥ 6, -q(G-)- ≤ | 16-, if n = 5, H (G ) || 5 |( 3, if n = 4, and the bounds are best possible.