scholarly journals Spectral characterizations of sun graphs and broken sun graphs

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Romain Boulet
Keyword(s):  
Graph Theory ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Difficult Problem ◽  
Laplacian Spectrum ◽  
Pendant Vertex ◽  
Unicyclic Graphs ◽  
Adjacency Spectrum ◽  
International Audience ◽  
Spectral Characterizations

International audience Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question ''Which graphs are determined by their spectrum?'' remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns.

scholarly journals Spectral Characterizations of Dumbbell Graphs

10.37236/314 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jianfeng Wang ◽  
Francesco Belardo ◽  
Qiongxiang Huang ◽  
Enzo M. Li Marzi
Keyword(s):  
Spectral Characterization ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Disjoint Cycles ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Unique Graph ◽  
Vertex Disjoint ◽  
Spectral Characterizations

A dumbbell graph, denoted by $D_{a,b,c}$, is a bicyclic graph consisting of two vertex-disjoint cycles $C_a$, $C_b$ and a path $P_{c+3}$ ($c \geq -1$) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of $D_{a,b,0}$ (without cycles $C_4$) with $\gcd(a,b)\geq 3$, and we complete the research started in [J.F. Wang et al., A note on the spectral characterization of dumbbell graphs, Linear Algebra Appl. 431 (2009) 1707–1714]. In particular we show that $D_{a,b,0}$ with $3 \leq \gcd(a,b) < a$ or $\gcd(a,b)=a$ and $b\neq 3a$ is determined by the spectrum. For $b=3a$, we determine the unique graph cospectral with $D_{a,3a,0}$. Furthermore we give the spectral characterization w.r.t. the signless Laplacian spectrum of all dumbbell graphs.

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.

CHARACTERISTIC POLYNOMIAL OF ADJACENCY OR LAPLACIAN MATRIX FOR WEIGHTED TREELIKE NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950074 ◽  
Author(s):  
MEIFENG DAI ◽  
YONGBO HOU ◽  
CHANGXI DAI ◽  
TINGTING JU ◽  
YU SUN ◽  
...  
Keyword(s):  
Characteristic Polynomial ◽  
Future Study ◽  
Laplacian Matrix ◽  
Block Matrix ◽  
Weight Factor ◽  
Laplacian Spectrum ◽  
Weighted Networks ◽  
Adjacency Spectrum ◽  
Analytic Expression ◽  
Successive Generations

In recent years, weighted networks have been extensively studied in various fields. This paper studies characteristic polynomial of adjacency or Laplacian matrix for weighted treelike networks. First, a class of weighted treelike networks with a weight factor is introduced. Then, the relationships of adjacency or the Laplacian matrix at two successive generations are obtained. Finally, according to the operation of the block matrix, we obtain the analytic expression of the characteristic polynomial of the adjacency or the Laplacian matrix. The obtained results lay the foundation for the future study of adjacency spectrum or Laplacian spectrum.

scholarly journals Adjacency Matrix of Product of Graphs

10.29007/fqlw ◽  
2018 ◽  
Author(s):  
Urvashi Acharya ◽  
Himali Mehta
Keyword(s):  
Graph Theory ◽  
Tensor Product ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Cartesian Product ◽  
Laplacian Matrix ◽  
New Product ◽  
Different Types ◽  
Product Of Graphs

In graph theory, different types of matrices associated with graph, e.g. Adjacency matrix, Incidence matrix, Laplacian matrix etc. Among all adjacency matrix play an important role in graph theory. Many products of two graphs as well as its generalized form had been studied, e.g., cartesian product, 2−cartesian product, tensor product, 2−tensor product etc. In this paper, we discuss the adjacency matrix of two new product of graphs G H, where = ⊗2, ×2. Also, we obtain the spectrum of these products of graphs.

Spectrum of graphs obtained by operations

2018 ◽  
Vol 13 (02) ◽  
pp. 2050045
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Somnath Paul
Keyword(s):  
Distance Matrix ◽  
Laplacian Matrix ◽  
Regular Graphs ◽  
Lexicographic Product ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Simple Connected Graph ◽  
Equienergetic Graphs

The distance signless Laplacian matrix of a simple 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 diagonal entries are the vertex transmissions in [Formula: see text]. In this paper, we first determine the distance signless Laplacian spectrum of the graphs obtained by generalization of the join and lexicographic product graph operations (namely joined union) in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix, determined by the graph [Formula: see text]. As an application, we show that new pairs of auxiliary equienergetic graphs can be constructed by joined union of regular graphs.

scholarly journals The least Laplacian eigenvalue of the unbalanced unicyclic signed graphs with $k$ pendant vertices

2020 ◽  
Vol 36 (36) ◽  
pp. 390-399
Author(s):  
Qiao Guo ◽  
Yaoping Hou ◽  
Deqiong Li
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Signed Graph ◽  
Signed Graphs ◽  
Laplacian Eigenvalue ◽  
Unicyclic Graphs ◽  
Underlying Graph ◽  
Vertex Degrees

Let $\Gamma=(G,\sigma)$ be a signed graph and $L(\Gamma)=D(G)-A(\Gamma)$ be the Laplacian matrix of $\Gamma$, where $D(G)$ is the diagonal matrix of vertex degrees of the underlying graph $G$ and $A(\Gamma)$ is the adjacency matrix of $\Gamma$. It is well-known that the least Laplacian eigenvalue $\lambda_n$ is positive if and only if $\Gamma$ is unbalanced. In this paper, the unique signed graph (up to switching equivalence) which minimizes the least Laplacian eigenvalue among unbalanced connected signed unicyclic graphs with $n$ vertices and $k$ pendant vertices is 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.

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

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.

Sign in / Sign up

Export Citation Format

Share Document