scholarly journals On the Spectra of General Random Mixed Graphs

10.37236/9638 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Dan Hu ◽  
Hajo Broersma ◽  
Jiangyou Hou ◽  
Shenggui Zhang
Keyword(s):  
Error Bounds ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Laplacian Matrix ◽  
Probability Inequality ◽  
Mixed Graph ◽  
Mixed Graphs

A mixed graph is a graph that can be obtained from a simple undirected graph by replacing some of the edges by arcs in precisely one of the two possible directions. The Hermitian adjacency matrix of a mixed graph $G$ of order $n$ is the $n \times n$ matrix $H(G)=(h_{ij})$, where $h_{ij}=-h_{ji}= \boldsymbol{\mathrm{i}}$ (with $\boldsymbol{\mathrm{i}} =\sqrt{-1})$ if there exists an arc from $v_i$ to $v_j$ (but no arc from $v_j$ to $v_i$), $h_{ij}=h_{ji}=1$ if there exists an edge (and no arcs) between $v_i$ and $v_j$, and $h_{ij}= 0$ otherwise (if $v_i$ and $v_j$ are neither joined by an edge nor by an arc). We study the spectra of the Hermitian adjacency matrix and the normalized Hermitian Laplacian matrix of general random mixed graphs, i.e., in which all arcs are chosen independently with different probabilities (and an edge is regarded as two oppositely oriented arcs joining the same pair of vertices). For our first main result, we derive a new probability inequality and apply it to obtain an upper bound on the eigenvalues of the Hermitian adjacency matrix. Our second main result shows that the eigenvalues of the normalized Hermitian Laplacian matrix can be approximated by the eigenvalues of a closely related weighted expectation matrix, with error bounds depending on the minimum expected degree of the underlying undirected graph.

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.

scholarly journals Signless Laplacian determinations of some graphs with independent edges

2018 ◽  
Vol 10 (1) ◽  
pp. 185-196 ◽  
Author(s):  
R. Sharafdini ◽  
A.Z. Abdian
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Degree Matrix

Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.

scholarly journals On Graphs whose Orientations are Determined by their Hermitian Spectra

10.37236/9640 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Yi Wang ◽  
Bo-Jun Yuan
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Simple Graph ◽  
Equivalence Classes ◽  
Cycle Space ◽  
Mixed Graph ◽  
Cycle Basis ◽  
Underlying Graph ◽  
Mixed Graphs ◽  
Special Cycle

A mixed graph $D$ is obtained from a simple graph $G$, the underlying graph of $D$, by orienting some edges of $G$. A simple graph $G$ is said to be ODHS (all orientations of $G$ are determined by their $H$-spectra) if every two $H$-cospectral graphs in $\mathcal{D}(G)$ are switching equivalent to each other, where $\mathcal{D}(G)$ is the set of all mixed graphs with $G$ as their underlying graph. In this paper, we characterize all bicyclic ODHS graphs and construct infinitely many ODHS graphs whose cycle spaces are of dimension $k$. For a  connected graph $G$ whose cycle space is of dimension $k$, we also obtain an achievable upper bound $2^{2k-1} + 2^{k-1}$ for the number of switching equivalence classes in $\mathcal{D}(G)$, which naturally is an upper bound of the number of  cospectral classes in $\mathcal{D}(G)$. To achieve these, we propose a valid method to estimate the number of switching equivalence classes in $\mathcal{D}(G)$ based on the strong cycle basis, a special cycle basis  introduced in this paper.

On the Laplacian characteristic polynomials of mixed graphs

2015 ◽  
Vol 30 ◽  
Author(s):  
Dariush Kiani ◽  
Maryam Mirzakhah
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Characteristic Polynomials ◽  
Laplacian Spectral Radius ◽  
Mixed Graph ◽  
First Derivative ◽  
Signless Laplacian ◽  
Mixed Graphs

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial of a mixed graph is reconstructible from the collection of the Laplacian characteristic polynomials of its edge deleted subgraphs. Then, it is investigated how graph modifications affect the mixed Laplacian characteristic polynomial. Also, a connection between the Laplacian characteristic polynomial of a non-singular connected mixed graph and the signless Laplacian characteristic polynomial is provided, and it is used to establish a lower bound for the spectral radius of L(G). Finally, using Coates digraphs, the perturbation of the mixed Laplacian spectral radius under some graph transformations is discussed.

scholarly journals The p-spectral radius of the Laplacian matrix

2018 ◽  
Vol 12 (2) ◽  
pp. 455-466
Author(s):  
Elizandro Borba ◽  
Eliseu Fritscher ◽  
Carlos Hoppen ◽  
Sebastian Richter
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Laplacian Matrix ◽  
Extremal Problems ◽  
Graph Invariants ◽  
Maximum Cut

The p-spectral radius of a graph G=(V,E) with adjacency matrix A is defined as ?(p)(G) = max||x||p=1 xT Ax. This parameter shows connections with graph invariants, and has been used to generalize some extremal problems. In this work, we define the p-spectral radius of the Laplacian matrix L as ?(p)(G) = max||x||p=1 xT Lx. We show that ?(p)(G) relates to invariants such as maximum degree and size of a maximum cut. We also show properties of ?(p)(G) as a function of p, and a upper bound on maxG: |V(G)|=n ?(p)(G) in terms of n = |V| for p > 2, which is attained if n is even.

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.

Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs

2017 ◽  
Vol 293 ◽  
pp. 287-292 ◽  
Author(s):  
Guihai Yu ◽  
Xin Liu ◽  
Hui Qu
Keyword(s):  
Laplacian Matrix ◽  
Mixed Graphs

scholarly journals Spectral property of certain class of graphs associated with generalized Bethe trees and transitive graphs

2008 ◽  
Vol 2 (2) ◽  
pp. 260-275 ◽  
Author(s):  
Yi-Zheng Fan ◽  
Li Shuang-Dong ◽  
Dong Liang
Keyword(s):  
Spectral Property ◽  
Adjacency Matrix ◽  
Rooted Tree ◽  
Laplacian Matrix ◽  
Transitive Graph ◽  
Tridiagonal Matrices ◽  
Equal Degree ◽  
Transitive Graphs ◽  
Structure Properties

A generalized Bethe tree is a rooted tree for which the vertices in each level having equal degree. Let Bk be a generalized Bethe tree of k level, and let T r be a connected transitive graph on r vertices. Then we obtain a graph Bk?T r from r copies of Bk and T r by appending r roots to the vertices of T r respectively. In this paper, we give a simple way to characterize the eigenvalues of the adjacency matrix A(Bk ? T r) and the Laplacian matrix L(Bk?T r) of Bk?T r by means of symmetric tridiagonal matrices of order k. We also present some structure properties of the Perron vectors of A(Bk?T r) and the Fiedler vectors of L(Bk ? T r). In addition, we obtain some results on transitive graphs.

scholarly journals Bounds for the Generalized Distance Eigenvalues of a Graph

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1529 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal Ahmad Ganie ◽  
Yilun Shang
Keyword(s):  
Spectral Radius ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Extremal Graphs ◽  
Special Cases ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Generalized Distance

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue.

scholarly journals The Unique Mixed Almost Moore Graph with Parameters k = 2, r = 2 and z = 1

2017 ◽  
Vol 17 (03n04) ◽  
pp. 1741005 ◽  
Author(s):  
DOMINIQUE BUSET ◽  
NACHO LÓPEZ ◽  
JOSEP M. MIRET
Keyword(s):  
Upper Bound ◽  
Mixed Graph ◽  
Moore Graph

A natural upper bound for the maximum number of vertices in a mixed graph with maximum undirected degree r, maximum directed out-degree z and diameter k is given by the mixed Moore bound. Graphs with order attaining the Moore bound are known as Moore graphs, and they are very rare. Besides, graphs with prescribed parameters and order one less than the corresponding Moore bound are known as almost Moore graphs. In this paper we prove that there is a unique mixed almost Moore graph of diameter k = 2 and parameters r = 2 and z = 1.

Sign in / Sign up

Export Citation Format

Share Document