scholarly journals Distance spectrum and energy of graphs with small diameter

2017 ◽  
Vol 11 (1) ◽  
pp. 108-122 ◽  
Author(s):  
Milica Andjelic ◽  
Tamara Koledin ◽  
Zoran Stanic
Keyword(s):  
Adjacency Matrix ◽  
Small Diameter ◽  
Distance Spectrum ◽  
Spectrum Of Graphs ◽  
Equienergetic Graphs

In this paper we express the distance spectrum of graphs with small diameter in terms of the eigenvalues of their adjacency matrix. We also compute the distance energy of particular types of graph and determine a sequence of infinite families of distance equienergetic graphs.

scholarly journals On the distance spectrum of graphs

2013 ◽  
Vol 439 (6) ◽  
pp. 1662-1669 ◽  
Author(s):  
Huiqiu Lin ◽  
Yuan Hong ◽  
Jianfeng Wang ◽  
Jinlong Shu
Keyword(s):  
Distance Spectrum ◽  
Spectrum Of Graphs

scholarly journals On revised szeged spectrum of a graph

2014 ◽  
Vol 45 (4) ◽  
pp. 375-387 ◽  
Author(s):  
Nader Habibi ◽  
Ali Reza Ashrafi
Keyword(s):  
Molecular Structure ◽  
Adjacency Matrix ◽  
Molecular Graph ◽  
Szeged Index ◽  
Sum Of Products ◽  
Structure Descriptor ◽  
Spectrum Of Graphs

The revised Szeged index is a molecular structure descriptor equal to the sum of products $[n_u(e) + \frac{n_0(e)}{2}][n_v(e) + \frac{n_0(e)}{2}]$ over all edges $e = uv$ of the molecular graph $G$, where $n_0(e)$ is the number of vertices equidistant from $u$ and $v$, $n_u(e)$ is the number of vertices closer to $u$ than $v$ and $n_v(e)$ is defined analogously. The adjacency matrix of a graph weighted in this way is called its revised Szeged matrix and the set of its eigenvalues is the revised Szeged spectrum of $G$. In this paper some new results on the revised Szeged spectrum of graphs are presented.

scholarly journals On Equienergetic, Hyperenergetic and Hypoenergetic Graphs

2020 ◽  
Vol 44 (4) ◽  
pp. 523-532
Author(s):  
SAMIR K. VAIDYA ◽  
KALPESH POPAT
Keyword(s):  
Complete Graph ◽  
Adjacency Matrix ◽  
Affirmative Answer ◽  
Natural Question ◽  
Isomorphic Graphs ◽  
Equienergetic Graphs

The eigenvalue of a graph G is the eigenvalue of its adjacency matrix and the energy E(G) is the sum of absolute values of eigenvalues of graph G. Two non-isomorphic graphs G1 and G2 of the same order are said to be equienergetic if E(G1) = E(G2). The graphs whose energy is greater than that of complete graph are called hyperenergetic and the graphs whose energy is less than that of its order are called hypoenergetic graphs. The natural question arises: Are there any pairs of equienergetic graphs which are also hyperenergetic (hypoenergetic)? We have found an affirmative answer of this question and contribute some new results.

Graphs with few distinct distance eigenvalues irrespective of the diameters

2015 ◽  
Vol 29 ◽  
pp. 194-205 ◽  
Author(s):  
Fouzul Atik ◽  
Pratima Panigrahi
Keyword(s):  
Distance Matrix ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Arbitrary Graph ◽  
Distance Spectrum ◽  
Strong Product ◽  
Strongly Regular ◽  
Distinct Distance ◽  
Simple Connected Graph ◽  
Spectrum Of Graphs

The distance matrix of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between $i$th and $j$th vertices of $G$. The multiset of all eigenvalues of $D(G)$ is known as the distance spectrum of $G$. Lin et al.(On the distance spectrum of graphs. \newblock {\em Linear Algebra Appl.}, 439:1662-1669, 2013) asked for existence of graphs other than strongly regular graphs and some complete $k$-partite graphs having exactly three distinct distance eigenvalues. In this paper some classes of graphs with arbitrary diameter and satisfying this property is constructed. For each $k\in \{4,5,\ldots,11\}$ families of graphs that contain graphs of each diameter grater than $k-1$ is constructed with the property that the distance matrix of each graph in the families has exactly $k$ distinct eigenvalues. While making these constructions we have found the full distance spectrum of square of even cycles, square of hypercubes, corona of a transmission regular graph with $K_2$, and strong product of an arbitrary graph with $K_n$

Energy of Graphs

2020 ◽  
pp. 267-296 ◽  
Author(s):  
Harishchandra S. Ramane
Keyword(s):  
Molecular Orbital ◽  
Electron Energy ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Close Correlation ◽  
Orbital Method ◽  
Molecular Orbital Method ◽  
Cospectral Graphs ◽  
Graph Energy ◽  
Equienergetic Graphs

The energy of a graph G is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. The graph energy has close correlation with the total pi-electron energy of molecules calculated with Huckel molecular orbital method in chemistry. A graph whose energy is greater than the energy of complete graph of same order is called hyperenergetic graph. A non-complete graph having energy equal to the energy of complete graph is called borderenergetic graph. Two non-cospectral graphs are said to be equienergetic graphs if they have same energy. In this chapter, the results on graph energy are reported. Various bounds for graph energy and its characterization are summarized. Construction of hyperenergetic, borderenergetic, and equienergetic graphs are reported.

A Replica Technique for the Inside Surfaces of Small Diameter Pipes and Tubes

1967 ◽  
Vol 25 ◽  
pp. 352-353
Author(s):  
T. G. Gregory
Keyword(s):  
Polyvinyl Chloride ◽  
Small Diameter ◽  
Replica Technique ◽  
Tube Size ◽  
Inside Diameter ◽  
Complete Inspection ◽  
Operating Limits

A nondestructive replica technique permitting complete inspection of bore surfaces having an inside diameter from 0.050 inch to 0.500 inch is described. Replicas are thermally formed on the outside surface of plastic tubing inflated in the bore of the sample being studied. This technique provides a new medium for inspection of bores that are too small or otherwise beyond the operating limits of conventional inspection methods.Bore replicas may be prepared by sliding a length of plastic tubing completely through the bore to be studied as shown in Figure 1. Polyvinyl chloride tubing suitable for this replica process is commercially available in sizes from 0.037- to 0.500-inch diameter. A tube size slightly smaller than the bore to be replicated should be used to facilitate insertion of the plastic replica blank into the bore.

Structural Organization and Distribution of Fiber Types in Extraocular Muscles of Cats

1972 ◽  
Vol 30 ◽  
pp. 34-35
Author(s):  
Asish C. Nag ◽  
Lee D. Peachey
Keyword(s):  
Fiber Type ◽  
Light And Electron Microscopy ◽  
Small Diameter ◽  
Extraocular Muscles ◽  
Fiber Types ◽  
Distinctive Features ◽  
Superior Rectus ◽  
Adult Cats ◽  
Different Diameters ◽  
Orbital Region

Cat extraocular muscles consist of two regions: orbital, and global. The orbital region contains predominantly small diameter fibers, while the global region contains a variety of fibers of different diameters. The differences in ultrastructural features among these muscle fibers indicate that the extraocular muscles of cats contain at least five structurally distinguishable types of fibers.Superior rectus muscles were studied by light and electron microscopy, mapping the distribution of each fiber type with its distinctive features. A mixture of 4% paraformaldehyde and 4% glutaraldehyde was perfused through the carotid arteries of anesthetized adult cats and applied locally to exposed superior rectus muscles during the perfusion.

Prospects for convergent beam electron diffraction with a 200kV cold field-emission transmission electron microscope

1991 ◽  
Vol 49 ◽  
pp. 466-467
Author(s):  
J W Steeds ◽  
R Vincent
Keyword(s):  
Field Emission ◽  
High Performance ◽  
Small Diameter ◽  
Convergent Beam Electron Diffraction ◽  
Zone Axis ◽  
Medium Voltage ◽  
Transmission Electron ◽  
Good Crystallinity ◽  
Special Modification ◽  
Convergent Beam

We review the analytical powers which will become more widely available as medium voltage (200-300kV) TEMs with facilities for CBED on a nanometre scale come onto the market. Of course, high performance cold field emission STEMs have now been in operation for about twenty years, but it is only in relatively few laboratories that special modification has permitted the performance of CBED experiments. Most notable amongst these pioneering projects is the work in Arizona by Cowley and Spence and, more recently, that in Cambridge by Rodenburg and McMullan.There are a large number of potential advantages of a high intensity, small diameter, focussed probe. We discuss first the advantages for probes larger than the projected unit cell of the crystal under investigation. In this situation we are able to perform CBED on local regions of good crystallinity. Zone axis patterns often contain information which is very sensitive to thickness changes as small as 5nm. In conventional CBED, with a lOnm source, it is very likely that the information will be degraded by thickness averaging within the illuminated area.

Applications of SIMS to electronic materials and devices

1992 ◽  
Vol 50 (2) ◽  
pp. 1682-1683
Author(s):  
S.F. Corcoran
Keyword(s):  
Depth Distribution ◽  
Secondary Ion Mass Spectrometry ◽  
Semiconductor Device ◽  
Electronic Materials ◽  
Profile Analysis ◽  
Semiconductor Industry ◽  
Small Diameter ◽  
Depth Resolution ◽  
Detection Sensitivity

Over the past decade secondary ion mass spectrometry (SIMS) has played an increasingly important role in the characterization of electronic materials and devices. The ability of SIMS to provide part per million detection sensitivity for most elements while maintaining excellent depth resolution has made this technique indispensable in the semiconductor industry. Today SIMS is used extensively in the characterization of dopant profiles, thin film analysis, and trace analysis in bulk materials. The SIMS technique also lends itself to 2-D and 3-D imaging via either the use of stigmatic ion optics or small diameter primary beams.By far the most common application of SIMS is the determination of the depth distribution of dopants (B, As, P) intentionally introduced into semiconductor materials via ion implantation or epitaxial growth. Such measurements are critical since the dopant concentration and depth distribution can seriously affect the performance of a semiconductor device. In a typical depth profile analysis, keV ion sputtering is used to remove successive layers the sample.

Improved immunolabelling of ultrathin cryosections using antibody conjugated with 1nm gold particles

1990 ◽  
Vol 48 (3) ◽  
pp. 928-929
Author(s):  
Morten H. Nielsen ◽  
Lone Bastholm
Keyword(s):  
Pituitary Gland ◽  
Liquid Nitrogen ◽  
Electron Density ◽  
Small Diameter ◽  
Cytochemical Staining ◽  
Gold Particles ◽  
Labelling Density ◽  
Ultrathin Cryosections ◽  
Mouse Pituitary ◽  
Electron Microscopical

During the last 5 years the diameter of the gold probes used for immuno-cytochemical staining at the electron microscopical (EM) level has been decreased. The advantage of small diameter gold probes is an overall increased labelling density. The disadvantage is a lower detectability due to the low electron density of smaller gold particles consequently an inconvenient high primary magnification needed for EM examination. Since 1 nm gold particles are barely visible by conventional EM examination the need for enlargement by silverenhancement of the gold particles has increased.In the present study of ultrathin cryosectioned material the results of immunostaining using 5 nm gold conjugated antibody and 1 nm gold conjugated antibodies are compared after silverenhancement of the 1 nm gold particles.Slices of freshly isolated mouse pituitary gland were immersion fixed for 20 min in 2 % glutaraldehyde /2 % paraformaldehyde. Blocks cryoprotected with 2.3 M sucrose were frozen in liquid nitrogen and ultra-cryosectioned on a RMC cryoultra-microtome.

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