scholarly journals The Optimal Graph Whose Least Eigenvalue is Minimal among All Graphs via 1-2 Adjacency Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Lubna Gul ◽  
Gohar Ali ◽  
Usama Waheed ◽  
Nudrat Aamir
Keyword(s):  
Adjacency Matrix ◽  
Least Eigenvalue ◽  
New Type ◽  
Unique Graph ◽  
Optimal Graph

All graphs under consideration are finite, simple, connected, and undirected. Adjacency matrix of a graph G is 0,1 matrix A = a i j = 0 ,   i f   v i = v j   o r     d v i , v j ≥ 2 1 ,   i f     d v i , v j = 1. . Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 G = a i j = 0 ,   i f   v i = v j   o r     d v i , v j ≥ 3 1 ,   i f     d v i , v j = 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T n c be the set of the complement of trees of order n . In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T n c .

scholarly journals The Least Eigenvalue of the Graphs Whose Complements Are Connected and Have Pendent Paths

2018 ◽  
Vol 8 (4) ◽  
pp. 303-308 ◽  
Author(s):  
Chen Wang ◽  
Guidong Yu ◽  
Wei Sun ◽  
Jinde Cao
Keyword(s):  
Adjacency Matrix ◽  
Minimum Eigenvalue ◽  
Least Eigenvalue ◽  
Unique Graph ◽  
Adjacent Relation

Abstract The adjacency matrix of a graph is a matrix which represents adjacent relation between the vertices of the graph. Its minimum eigenvalue is defined as the least eigenvalue of the graph. Let Gn be the set of the graphs of order n, whose complements are connected and have pendent paths. This paper investigates the least eigenvalue of the graphs and characterizes the unique graph which has the minimum least eigenvalue in Gn.

scholarly journals Least eigenvalue of the connected graphs whose complements are cacti

2019 ◽  
Vol 17 (1) ◽  
pp. 1319-1331
Author(s):  
Haiying Wang ◽  
Muhammad Javaid ◽  
Sana Akram ◽  
Muhammad Jamal ◽  
Shaohui Wang
Keyword(s):  
Linear Algebra ◽  
Adjacency Matrix ◽  
Connected Graphs ◽  
Least Eigenvalue ◽  
The Family

Abstract Suppose that Γ is a graph of order n and A(Γ) = [ai,j] is its adjacency matrix such that ai,j is equal to 1 if vi is adjacent to vj and ai,j is zero otherwise, where 1 ≤ i, j ≤ n. In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrović et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we characterize the minimizing graphs G ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, i.e. $$\begin{array}{} \displaystyle \lambda_{min}(G)\leq\lambda_{min}(C^c) \end{array}$$ for each Cc ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, where $\begin{array}{} {\it\Omega}^c_n \end{array}$ is a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle.

scholarly journals On the nullity of connected graphs with least eigenvalue at least -2

2013 ◽  
Vol 7 (2) ◽  
pp. 250-261 ◽  
Author(s):  
Jiang Zhou ◽  
Lizhu Sun ◽  
Hongmei Yao ◽  
Changjiang Bu
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Line Graph ◽  
Line Graphs ◽  
Connected Graphs ◽  
Least Eigenvalue

Let L (resp. L+) be the set of connected graphs with least adjacency eigenvalue at least -2 (resp. larger than -2). The nullity of a graph G, denoted by ?(G), is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. In this paper, we give the nullity set of L+ and an upper bound on the nullity of exceptional graphs. An expression for the nullity of generalized line graphs is given. For G ? L, if ?(G) is sufficiently large, then G is a proper generalized line graph (G is not a line graph).

scholarly journals Line graphs of complex unit gain graphs with least eigenvalue -2

2021 ◽  
Vol 37 (37) ◽  
pp. 14-30
Author(s):  
Maurizio Brunetti ◽  
Francesco Belardo
Keyword(s):  
Real Number ◽  
Adjacency Matrix ◽  
Multiplicative Group ◽  
Line Graph ◽  
Line Graphs ◽  
Signed Graphs ◽  
The Real ◽  
Least Eigenvalue ◽  
Gain Graphs ◽  
Gain Graph

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph of a $\mathbb{T}$-gain graph $\Phi$. Similarly to what happens in the context of signed graphs, the real number $\min Spec(A(\mathcal L (\Phi))$, that is, the smallest eigenvalue of the adjacency matrix of $\mathcal L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring that $\min Spec(A(\mathcal L (\Phi))=-2$ are identified. When such conditions are fulfilled, bases of the $-2$-eigenspace are constructed with the aid of the star complement technique.

On the Estrada index of cacti

2014 ◽  
Vol 27 ◽  
Author(s):  
Erfang Shan ◽  
Hongzhuan Wang ◽  
Liying Kang
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Common Vertex ◽  
Estrada Index ◽  
Unique Graph ◽  
Simple Connected Graph

Let G be a simple connected graph on n vertices and λ_1, λ_2, . . . , λ_n be the eigenvalues of the adjacency matrix of G. The Estrada index of G is defined as EE(G) = \sum_{i=1}^n e^{λi}. A cactus is a connected graph in which any two cycles have at most one common vertex. In this work, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cycles was determined. Also, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cut edges was determined.

The Microstructure of Shock-Synthesized Diamond

1967 ◽  
Vol 25 ◽  
pp. 324-325
Author(s):  
Lucien F. Trueb
Keyword(s):  
Single Crystals ◽  
Preferred Orientation ◽  
High Magnification ◽  
Texture Pattern ◽  
Shock Process ◽  
New Type ◽  
Diffraction Diagram

A new type of synthetic industrial diamond formed by an explosive shock process has been recently developed by the Du Pont Company. This material consists of a mixture of two basically different forms, as shown in Figure 1: relatively flat and compact aggregates of acicular crystallites, and single crystals in the form of irregular polyhedra with straight edges.Figure 2 is a high magnification micrograph typical for the fibrous aggregates; it shows that they are composed of bundles of crystallites 0.05-0.3 μ long and 0.02 μ. wide. The selected area diffraction diagram (insert in Figure 2) consists of a weak polycrystalline ring pattern and a strong texture pattern with arc reflections. The latter results from crystals having preferred orientation, which shows that in a given particle most fibrils have a similar orientation.

New type electron gun with electrostatic and electromagnetic lenses

1978 ◽  
Vol 36 (1) ◽  
pp. 62-63
Author(s):  
T. Ichinokawa ◽  
H. Maeda
Keyword(s):  
Electron Microscopy ◽  
Electron Beam ◽  
Optimum Condition ◽  
Bias Voltage ◽  
Electron Gun ◽  
Charge Effect ◽  
Micro Fabrication ◽  
Maximum Brightness ◽  
New Type ◽  
The Ideal

I. IntroductionThermionic electron gun with the Wehnelt grid is popularly used in the electron microscopy and electron beam micro-fabrication. It is well known that this gun could get the ideal brightness caluculated from the Lengumier and Richardson equations under the optimum condition. However, the design and ajustment to the optimum condition is not so easy. The gun has following properties with respect to the Wehnelt bias; (1) The maximum brightness is got only in the optimum bias. (2) In the larger bias than the optimum, the brightness decreases with increasing the bias voltage on account of the space charge effect. (3) In the smaller bias than the optimum, the brightness decreases with bias voltage on account of spreading of the cross over spot due to the aberrations of the electrostatic immersion lens.In the present experiment, a new type electron gun with the electrostatic and electromagnetic lens is designed, and its properties are examined experimentally.

A new type of defect structure in 124-superconducting oxide revealed by HREM

1991 ◽  
Vol 49 ◽  
pp. 1092-1093
Author(s):  
R. Sharma ◽  
B.L. Ramakrishna ◽  
N.N. Thadhani ◽  
D. Hianes ◽  
Z. Iqbal
Keyword(s):  
Shock Wave ◽  
Defect Structure ◽  
Neutron Radiation ◽  
Weak Links ◽  
Defect Structures ◽  
New Materials ◽  
New Type ◽  
Current Densities ◽  
Processing Techniques ◽  
Microstructural Studies

After materials with superconducting temperatures higher than liquid nitrogen have been prepared, more emphasis has been on increasing the current densities (Jc) of high Tc superconductors than finding new materials with higher transition temperatures. Different processing techniques i.e thin films, shock wave processing, neutron radiation etc. have been applied in order to increase Jc. Microstructural studies of compounds thus prepared have shown either a decrease in gram boundaries that act as weak-links or increase in defect structure that act as flux-pinning centers. We have studied shock wave synthesized Tl-Ba-Cu-O and shock wave processed Y-123 superconductors with somewhat different properties compared to those prepared by solid-state reaction. Here we report the defect structures observed in the shock-processed Y-124 superconductors.

ESEM - A multipurpose surface Electron Microscope

1986 ◽  
Vol 44 ◽  
pp. 632-633 ◽  
Author(s):  
G.D. Danilatos
Keyword(s):  
Electron Microscope ◽  
Room Temperature ◽  
Environmental Scanning ◽  
Gas Composition ◽  
Saturated Water Vapor ◽  
Surface Electron ◽  
Current State ◽  
Saturated Water ◽  
New Type ◽  
Temperature And Pressure

Over recent years a new type of electron microscope - the environmental scanning electron microscope (ESEM) - has been developed for the examination of specimen surfaces in the presence of gases. A detailed series of reports on the system has appeared elsewhere. A review summary of the current state and potential of the system is presented here.The gas composition, temperature and pressure can be varied in the specimen chamber of the ESEM. With air, the pressure can be up to one atmosphere (about 1000 mbar). Environments with fully saturated water vapor only at room temperature (20-30 mbar) can be easily maintained whilst liquid water or other solutions, together with uncoated specimens, can be imaged routinely during various applications.

Trial production of γ-type energy filtering TEM

1995 ◽  
Vol 53 ◽  
pp. 604-605
Author(s):  
Y. Taniguchi ◽  
E. Nakazawa ◽  
S. Taya
Keyword(s):  
Magnetic Energy ◽  
Electron Optics ◽  
Electron Microscopic ◽  
Mass Spectrometers ◽  
Ion Optics ◽  
Energy Filtering ◽  
New Information ◽  
New Type ◽  
Fringing Fields

Imaging energy filters can add new information to electron microscopic images with respect to energy-axis, so-called electron spectroscopic imaging (ESI). Recently, many good results have been reported using this imaging technique. ESI also allows high-contrast observation of unstained biological samples, becoming a trend of the field of morphology. We manufactured a new type of energy filter as a trial production. This energy filter consists of two magnets, and we call γ-filter since the trajectory of electrons shows ‘γ’-shape inside the filter. We evaluated the new energyγ-filter TEM with the γ-filter.Figure 1 shows schematic view of the electron optics of the γ-type energy filter. For the determination of the electron-optics of the γ-type energy filter, we used the TRIO (Third Order Ion Optics) program which has been developed for the design of high resolution mass spectrometers. The TRIO takes the extended fringing fields (EFF) into consideration. EFF makes it difficult to design magnetic energy filters with magnetic sector fields.

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