scholarly journals Which non-regular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues?

2004 ◽  
Vol 286 (1-2) ◽  
pp. 15-24 ◽  
Author(s):  
Krystyna T. Balińska ◽  
Slobodan K. Simić ◽  
Krzysztof T. Zwierzyński
Keyword(s):  
Maximum Degree ◽  
Integral Graphs

scholarly journals Laplacian Integral Graphs with Maximum Degree $3$

10.37236/844 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Steve Kirkland
Keyword(s):  
Maximum Degree ◽  
Laplacian Matrix ◽  
Integral Graphs

A graph is said to be Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. Using combinatorial and matrix-theoretic techniques, we identify, up to isomorphism, the $21$ connected Laplacian integral graphs of maximum degree $3$ on at least $6$ vertices.

scholarly journals The nonregular, bipartite, integral graphs with maximum degree 4. Part I: basic properties

2001 ◽  
Vol 236 (1-3) ◽  
pp. 13-24 ◽  
Author(s):  
Krystyna T. Balińska ◽  
Slobodan K. Simić
Keyword(s):  
Maximum Degree ◽  
Basic Properties ◽  
Integral Graphs

scholarly journals More on non-regular bipartite integral graphs with maximum degree 4 not having ±1 as eigenvalues

2014 ◽  
Vol 8 (1) ◽  
pp. 123-154 ◽  
Author(s):  
Abreu de ◽  
Krystyna Balińska ◽  
Slobodan Simic ◽  
Krzysztof Zwierzyński
Keyword(s):  
Adjacency Matrix ◽  
Maximum Degree ◽  
Complete Solution ◽  
Discrete Math ◽  
Integral Graphs

A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ?1 as eigenvalues was posed in K.T. Bali?ska, S.K. Simic, K.T. Zwierzy?ski: Which non-regular bipartite integral graphs with maximum degree four do not have ?1 as eigenvalues?, Discrete Math., 286 (2004), 15{25. Here we revisit this problem, and provide its complete solution using mostly the theoretical arguments.

Non-standard methods for converting numbers from one number system to another

2020 ◽  
Vol 1 (9) ◽  
pp. 28-30
Author(s):  
D. M. Zlatopolski
Keyword(s):  
Binary System ◽  
Maximum Degree ◽  
Number System ◽  
Binary Form ◽  
Binary Number ◽  
Standard Methods ◽  
Optimal Variant ◽  
Decimal System ◽  
Large Numbers ◽  
Independent Work

The article describes a number of little-known methods for translating natural numbers from one number system to another. The first is a method for converting large numbers from the decimal system to the binary system, based on multiple divisions of a given number and all intermediate quotients by 64 (or another number equal to 2n ), followed by writing the last quotient and the resulting remainders in binary form. Then two methods of mutual translation of decimal and binary numbers are described, based on the so-called «Horner scheme». An optimal variant of converting numbers into the binary number system by the method of division by 2 is also given. In conclusion, a fragment of a manuscript from the beginning of the late 16th — early 17th centuries is published with translation into the binary system by the method of highlighting the maximum degree of number 2. Assignments for independent work of students are offered.

The Image of the Accountant in a German Context

2004 ◽  
Vol 4 (1) ◽  
pp. 62-89 ◽  
Author(s):  
Andreas Hoffjan
Keyword(s):  
Content Analysis ◽  
Empirical Study ◽  
Cost Reduction ◽  
Maximum Degree ◽  
Work Ethic ◽  
Personal Characteristics ◽  
Management Accountant ◽  
The Subject ◽  
Professional Context

This study introduces content analysis as a method of examining the accountant's role. The empirical study is based on 73 advertisements, which are directed primarily at employees who are affected by the management accountant's work. The findings of the study indicate that the subject of accountancy is used particularly in connection with promises of “cost reduction.” Consequently, the majority of advertisements use the accountant stereotype of “savings personified.” In a professional context, the work ethic of the management accountant is given particular emphasis in the advertisements. He/she identifies him/herself with his/her task to the maximum degree, is regarded as loyal to his/her company and, for the most part, is well organized in his/her work. However, the characterization of the management accountant as a well disciplined company-person conflicts with the negative portrayal of his/her professional qualities. In advertisements, the management accountant is portrayed as a rather inflexible, passive, and uncreative specialist who, as a result of these qualities, often demotivates others. The personal characteristics of the management accountant are shown in a negative light. This gives him/her the unappealing image of a humorless, envious, dissociated, and ascetic corporate-person.

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

scholarly journals The Complexity of Approximating the Matching Polynomial in the Complex Plane

2021 ◽  
Vol 13 (2) ◽  
pp. 1-37
Author(s):  
Ivona Bezáková ◽  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
Daniel Štefankovič
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Maximum Degree ◽  
Negative Real Axis ◽  
Positive Real ◽  
Bounded Degree ◽  
Matching Polynomial ◽  
Correlation Decay ◽  
Connective Constant ◽  
General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

Effects of hydrostatic pressure and inert gases on platelet aggregation in vitro

1990 ◽  
Vol 69 (6) ◽  
pp. 2239-2247 ◽  
Author(s):  
D. M. Pickles ◽  
D. Ogston ◽  
A. G. Macdonald
Keyword(s):  
Platelet Aggregation ◽  
Hydrostatic Pressure ◽  
Maximum Degree ◽  
Platelet Rich Plasma ◽  
Control Level ◽  
Inert Gases ◽  
Partial Pressures ◽  
Compression Cycle ◽  
Anesthetic Potency

A novel cuvette was used to subject citrated platelet-rich plasma (PRP) to high hydrostatic pressure with negligible contamination by He (used for compression of the apparatus). Aggregation was induced at pressure by ADP and quantified turbidimetrically. The maximum degree of aggregation (MDA) was reduced from a control level of 82.2 to 53.6% by exposure to 101 ATA. Because decompression bubbles did not form, aggregation was also measured immediately after a compression cycle. After exposure to 101 ATA hydrostatic pressure, platelets responded normally to ADP at 1 ATA. In a matching apparatus, PRP was equilibrated with high partial pressures of inert gases. Normal physiological plasma Po2 and pH were maintained during equilibration. N2O (5 ATA) reduced the MDA from 86.5 (control) to 58.1%. N2 (51 ATA) reduced the MDA from 74.7 (control) to 51.6%, and 101 ATA Pn2 reduced the MDA from 78.0 (control) to 32.3%. He (100 ATA) reduced the MDA from 83.6 to 38.6%. It was concluded that platelet aggregation was relatively sensitive to hydrostatic pressure and less sensitive to inert gases than predicted from their anesthetic potency ratios.

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