Order unicyclic graphs according to spetral radius of unoriented Laplacian matrix

2008 ◽  
Vol 28 (3) ◽  
pp. 487
Author(s):  
Yi-Zheng Fan ◽  
Song Wu
Keyword(s):  
Laplacian Matrix ◽  
Unicyclic Graphs

scholarly journals The Orderings of Bicyclic Graphs and Connected Graphs by Algebraic Connectivity

10.37236/434 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jianxi Li ◽  
Ji-Ming Guo ◽  
Wai Chee Shiu
Keyword(s):  
Linear Algebra ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Smallest Eigenvalue ◽  
Bicyclic Graphs

The algebraic connectivity of a graph $G$ is the second smallest eigenvalue of its Laplacian matrix. Let $\mathscr{B}_n$ be the set of all bicyclic graphs of order $n$. In this paper, we determine the last four bicyclic graphs (according to their smallest algebraic connectivities) among all graphs in $\mathscr{B}_n$ when $n\geq 13$. This result, together with our previous results on trees and unicyclic graphs, can be used to further determine the last sixteen graphs among all connected graphs of order $n$. This extends the results of Shao et al. [The ordering of trees and connected graphs by their algebraic connectivity, Linear Algebra Appl. 428 (2008) 1421-1438].

scholarly journals On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs withnVertices andmPendent Vertices

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Xinying Pai ◽  
Sanyang Liu
Keyword(s):  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Unicyclic Graphs ◽  
Pendent Vertices

LetΦ(G,λ)=det(λIn-L(G))=∑k=0n(-1)kck(G)λn-kbe the characteristic polynomial of the Laplacian matrix of a graphGof ordern. In this paper, we give four transforms on graphs that decrease all Laplacian coefficientsck(G)and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs withnvertices andmpendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs withnvertices andmpendent vertices.

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 The Least Algebraic Connectivity of Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Guisheng Jiang ◽  
Guidong Yu ◽  
Jinde Cao
Keyword(s):  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Unicyclic Graphs ◽  
Smallest Eigenvalue

The algebraic connectivity of a graph is defined as the second smallest eigenvalue of the Laplacian matrix of the graph, which is a parameter to measure how well a graph is connected. In this paper, we present two unique graphs whose algebraic connectivity attain the minimum among all graphs whose complements are trees, but not stars, and among all graphs whose complements are unicyclic graphs, but not stars adding one edge, respectively.

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 Identification of effective spreaders in contact networks using dynamical influence

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Ruaridh A. Clark ◽  
Malcolm Macdonald
Keyword(s):  
Disease Transmission ◽  
Average Rate ◽  
Laplacian Matrix ◽  
Disease Spread ◽  
Eigenvector Centrality ◽  
Contact Networks ◽  
Human Contact ◽  
Spread Dynamics ◽  
Definition Of ◽  
High Degree

AbstractContact networks provide insights on disease spread due to the duration of close proximity interactions. For systems governed by consensus dynamics, network structure is key to optimising the spread of information. For disease spread over contact networks, the structure would be expected to be similarly influential. However, metrics that are essentially agnostic to the network’s structure, such as weighted degree (strength) centrality and its variants, perform near-optimally in selecting effective spreaders. These degree-based metrics outperform eigenvector centrality, despite disease spread over a network being a random walk process. This paper improves eigenvector-based spreader selection by introducing the non-linear relationship between contact time and the probability of disease transmission into the assessment of network dynamics. This approximation of disease spread dynamics is achieved by altering the Laplacian matrix, which in turn highlights why nodes with a high degree are such influential disease spreaders. From this approach, a trichotomy emerges on the definition of an effective spreader where, for susceptible-infected simulations, eigenvector-based selections can either optimise the initial rate of infection, the average rate of infection, or produce the fastest time to full infection of the network. Simulated and real-world human contact networks are examined, with insights also drawn on the effective adaptation of ant colony contact networks to reduce pathogen spread and protect the queen ant.

scholarly journals Direct 3D model extraction method for color volume images

2021 ◽  
Vol 29 ◽  
pp. 133-140
Author(s):  
Bin Liu ◽  
Shujun Liu ◽  
Guanning Shang ◽  
Yanjie Chen ◽  
Qifeng Wang ◽  
...  
Keyword(s):  
3D Model ◽  
Clinical Medicine ◽  
Three Dimensional ◽  
Laplacian Matrix ◽  
Segmentation Method ◽  
Human Organs ◽  
Model Segmentation ◽  
Treatment Objective ◽  
Organ Models ◽  
Different Parts

BACKGROUND: There is a great demand for the extraction of organ models from three-dimensional (3D) medical images in clinical medicine diagnosis and treatment. OBJECTIVE: We aimed to aid doctors in seeing the real shape of human organs more clearly and vividly. METHODS: The method uses the minimum eigenvectors of Laplacian matrix to automatically calculate a group of basic matting components that can properly define the volume image. These matting components can then be used to build foreground images with the help of a few user marks. RESULTS: We propose a direct 3D model segmentation method for volume images. This is a process of extracting foreground objects from volume images and estimating the opacity of the voxels covered by the objects. CONCLUSIONS: The results of segmentation experiments on different parts of human body prove the applicability of this method.

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.

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