scholarly journals On the Hosoya Indices of Bicyclic Graphs with Small Diameter

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Tingzeng Wu ◽  
Yong Yu
Keyword(s):  
Small Diameter ◽  
Sharp Bound ◽  
Hosoya Index ◽  
Np Complete ◽  
Bicyclic Graphs

Let G be a graph. The Hosoya index of G , denoted by z G , is defined as the total number of its matchings. The computation of z G is NP-Complete. Wagner and Gutman pointed out that it is difficult to obtain results of the maximum Hosoya index among tree-like graphs with given diameter. In this paper, we focus on the problem, and a sharp bound of Hosoya indices of all bicyclic graphs with diameter of 3 is determined.

THE HOSOYA INDEX OF GRAPHS FORMED BY A FRACTAL GRAPH

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950135 ◽  
Author(s):  
JIA-BAO LIU ◽  
JING ZHAO ◽  
JIE MIN ◽  
JINDE CAO
Keyword(s):  
Computational Complexity ◽  
Hosoya Index ◽  
Initial Graph ◽  
Np Complete

The computational complexity of the Hosoya index of a given graph is NP-Complete. Let [Formula: see text] be the graph constructed from [Formula: see text] by a triangle instead of all vertices of the initial graph [Formula: see text]. In this paper, we characterize the Hosoya index of the graph [Formula: see text]. To our surprise, it shows that the Hosoya index of [Formula: see text] is thoroughly given by the order and degrees of all the vertices of the initial graph [Formula: see text].

scholarly journals Coloring Problems on Bipartite Graphs of Small Diameter

10.37236/9931 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Victor A. Campos ◽  
Guilherme C.M. Gomes ◽  
Allen Ibiapina ◽  
Raul Lopes ◽  
Ignasi Sau ◽  
...  
Keyword(s):  
Small Diameter ◽  
Bipartite Graphs ◽  
List Coloring ◽  
Coloring Problem ◽  
Np Completeness ◽  
Np Complete

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving $\textsf{NP}$-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$. Some of these results are obtained $\textsc{through}$ a proof that the Surjective $C_6$-Homomorphism problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vikas, 2017], we present ours as an alternative simpler one. As a byproduct, we also get that $3$-Biclique Partition is $\textsf{NP}$-complete. An attempt to prove this result was presented in [Fleischner, Mujuni, Paulusma, and Szeider, 2009], but there was a flaw in their proof, which we identify and discuss here. Finally, we prove that the $3$-Fall Coloring problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four, and prove that $\textsf{NP}$-completeness for diameter three would also imply $\textsf{NP}$-completeness of $3$-Precoloring Extension on diameter three, thus closing the previously mentioned open cases. This would also answer a question posed in [Kratochvíl, Tuza, and Voigt, 2002].

Maximum incomplete recursive circulants in graph embeddings

2015 ◽  
Vol 07 (04) ◽  
pp. 1550053
Author(s):  
R. Sundara Rajan ◽  
Indra Rajasingh ◽  
Paul Manuel ◽  
Mirka Miller ◽  
T. M. Rajalaxmi
Keyword(s):  
Parallel Algorithms ◽  
Small Diameter ◽  
Performance Degradation ◽  
Binary Trees ◽  
Graph Embeddings ◽  
Binomial Trees ◽  
Np Complete ◽  
Good Support ◽  
Special Classes

An incomplete recursive circulant possesses virtually every advantage of a complete recursive circulant, including simple deadlock-free routing, a small diameter, a good support of parallel algorithms, and so on. It is natural to reconfigure a faulty recursive circulant into a maximum incomplete recursive circulant so as to lower potential performance degradation. For [Formula: see text], the maximum incomplete subgraph problem is to identify a subgraph [Formula: see text] of a graph [Formula: see text] on [Formula: see text] vertices having the maximum number of edges among all subgraphs on [Formula: see text] vertices and is NP-complete. In this paper we identify maximum incomplete recursive circulants and use them as a tool to compute the exact wirelength of embedding recursive circulants into special classes of trees, such as [Formula: see text]-rooted complete binary trees, [Formula: see text]-rooted sibling trees, binomial trees, certain caterpillars and path.

scholarly journals The Maximum Hosoya Index of Unicyclic Graphs with Diameter at Most Four

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1034 ◽  
Author(s):  
Weijun Liu ◽  
Jingwen Ban ◽  
Lihua Feng ◽  
Tao Cheng ◽  
Frank Emmert-Streib ◽  
...  
Keyword(s):  
Small Diameter ◽  
Hosoya Index ◽  
Unicyclic Graphs

The Hosoya index of a graph is defined by the total number of the matchings of the graph. In this paper, we determine the maximum Hosoya index of unicyclic graphs with n vertices and diameter 3 or 4. Our results somewhat answer a question proposed by Wagner and Gutman in 2010 for unicyclic graphs with small diameter.

scholarly journals Complexity aspects of the computation of the rank of a graph

2014 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Igor Ramos ◽  
Vinícius F. Santos ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Convex Set ◽  
Small Diameter ◽  
Independent Set ◽  
Special Issue ◽  
Undirected Graphs ◽  
Radon Number ◽  
International Audience ◽  
Np Complete

Special issue PRIMA 2013 International audience We consider the P₃-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v \not ∈ H(S\setminus \v\) for all v in S. The rank \rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining \rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine \rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for \rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schäfer. Additionally, we briefly show that the problem is NP-complete also in the monophonic convexity.

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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