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

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 ◽  
Laplacian Spectral Radius ◽  
Harmonic Index ◽  
Signless Laplacian Spectral Radius ◽  
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.

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

2011 ◽  
Vol 03 (02) ◽  
pp. 185-191 ◽  
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 ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
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.

Further results on the distance signless Laplacian spectrum of graphs

2018 ◽  
Vol 11 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Ebrahim Hashemi
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Graph Parameters

The distance signless Laplacian matrix [Formula: see text] of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main entries are the vertex transmissions of [Formula: see text], and the spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of [Formula: see text]. In this paper, first we obtain the [Formula: see text]-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signless Laplacian spectral radius of a graph [Formula: see text] in terms of graph parameters and characterize the extremal graphs. Utilizing these results we present some upper and lower bounds on the distance signless Laplacian energy of a graph [Formula: see text].

On the distance signless Laplacian spectral radius and the distance signless Laplacian energy of graphs

2018 ◽  
Vol 10 (03) ◽  
pp. 1850035 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Somnath Paul
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Extremal Graphs ◽  
Graph Invariants ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Laplacian Energy ◽  
Signless Laplacian

The distance signless Laplacian spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of the distance signless Laplacian matrix of [Formula: see text], defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix of vertex transmissions of [Formula: see text]. In this paper, we determine some bounds on the distance signless Laplacian spectral radius of [Formula: see text] based on some graph invariants, and characterize the extremal graphs. In addition, we define distance signless Laplacian energy, similar to that in [J. Yang, L. You and I. Gutman, Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math. 37 (2013) 245–255] and give some bounds on the distance signless Laplacian energy of graphs.

scholarly journals On Distance Signless Laplacian Spectral Radius and Distance Signless Laplacian Energy

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 792
Author(s):  
Luis Medina ◽  
Hans Nina ◽  
Macarena Trigo
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Laplacian Energy ◽  
Signless Laplacian

In this article, we find sharp lower bounds for the spectral radius of the distance signless Laplacian matrix of a simple undirected connected graph and we apply these results to obtain sharp upper bounds for the distance signless Laplacian energy graph. The graphs for which those bounds are attained are characterized.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

On the distance signless Laplacian spectral radius of graphs and digraphs

2017 ◽  
Vol 32 ◽  
pp. 438-446 ◽  
Author(s):  
Dan Li ◽  
Guoping Wang ◽  
Jixiang Meng
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Laplacian Spectral Radius ◽  
Connected Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc

2018 ◽  
Vol 34 ◽  
pp. 191-204 ◽  
Author(s):  
Fouzul Atik ◽  
Pratima Panigrahi
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Nonnegative Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Ger\^sgorin disc of the distance (respectively, distance signless Laplacian) matrix.

Bounds for the skew Laplacian spectral radius of oriented graphs

2019 ◽  
Vol 35 (1) ◽  
pp. 31-40 ◽  
Author(s):  
BILAL A. CHAT ◽  
◽  
HILAL A. GANIE ◽  
S. PIRZADA ◽  
◽  
...  
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Arbitrary Direction ◽  
Laplacian Spectral Radius ◽  
Underlying Graph ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

We consider the skew Laplacian matrix of a digraph −→G obtained by giving an arbitrary direction to the edges of a graph G having n vertices and m edges. We obtain an upper bound for the skew Laplacian spectral radius in terms of the adjacency and the signless Laplacian spectral radius of the underlying graph G. We also obtain upper bounds for the skew Laplacian spectral radius and skew spectral radius, in terms of various parameters associated with the structure of the digraph −→G and characterize the extremal graphs.

scholarly journals Spectral Conditions for a Graph to be Hamilton-Connected

2013 ◽  
Vol 336-338 ◽  
pp. 2329-2334 ◽  
Author(s):  
Gui Dong Yu ◽  
Yi Zheng Fan
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Laplacian Spectral Radius ◽  
Hamiltonian Paths ◽  
Signless Laplacian Spectral Radius ◽  
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.

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