scholarly journals On degree sets in k-partite graphs

2020 ◽  
Vol 12 (2) ◽  
pp. 251-259
Author(s):  
T. A. Naikoo ◽  
U. Samee ◽  
S. Pirzada ◽  
Bilal A. Rather
Keyword(s):  
Degree Set

Abstract The degree set of a k-partite graph is the set of distinct degrees of its vertices. We prove that every set of non-negative integers is a degree set of some k-partite graph.

scholarly journals Reconstruction of score sets

2014 ◽  
Vol 6 (2) ◽  
pp. 210-229
Author(s):  
Antal Iványi
Keyword(s):  
Polynomial Time ◽  
Constructive Proof ◽  
Construction Algorithm ◽  
Polynomial Time Algorithms ◽  
General Polynomial ◽  
Degree Set ◽  
New Algorithms

Abstract The score set of a tournament is defined as the set of its different outdegrees. In 1978 Reid [15] published the conjecture that for any set of nonnegative integers D there exists a tournament T whose degree set is D. Reid proved the conjecture for tournaments containing n = 1, 2, and 3 vertices. In 1986 Hager [4] published a constructive proof of the conjecture for n = 4 and 5 vertices. In 1989 Yao [18] presented an arithmetical proof of the conjecture, but general polynomial construction algorithm is not known. In [6] we described polynomial time algorithms which reconstruct the score sets containing only elements less than 7. In [5] we improved this bound to 9. In this paper we present and analyze new algorithms Hole-Map, Hole-Pairs, Hole-Max, Hole-Shift, Fill-All, Prefix-Deletion, and using them improve the above bound to 12, giving a constructive partial proof of Reid’s conjecture.

scholarly journals Full-State Linearization and Stabilization of SISO Markovian Jump Nonlinear Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Zhongwei Lin ◽  
Jizhen Liu ◽  
Yuguang Niu
Keyword(s):  
Nonlinear Systems ◽  
Control Design ◽  
Relative Degree ◽  
Backstepping Technique ◽  
Markovian Jump ◽  
Design Problems ◽  
Stabilizing Control ◽  
Full State ◽  
Degree Set ◽  
Coordinate Changes

This paper investigates the linearization and stabilizing control design problems for a class of SISO Markovian jump nonlinear systems. According to the proposed relative degree set definition, the system can be transformed into the canonical form through the appropriate coordinate changes followed with the Markovian switchings; that is, the system can be full-state linearized in every jump mode with respect to the relative degree setn,…,n. Then, a stabilizing control is designed through applying the backstepping technique, which guarantees the asymptotic stability of Markovian jump nonlinear systems. A numerical example is presented to illustrate the effectiveness of our results.

The solvability comes from a given set of character degrees

2016 ◽  
Vol 15 (09) ◽  
pp. 1650164 ◽  
Author(s):  
Farideh Shafiei ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Prime Power ◽  
Character Degree ◽  
Character Degrees ◽  
Degree Set ◽  
A Finite Group ◽  
Irreducible Character Degree

Let [Formula: see text] be a finite group and the irreducible character degree set of [Formula: see text] is contained in [Formula: see text], where [Formula: see text], and [Formula: see text] are distinct integers. We show that one of the following statements holds: [Formula: see text] is solvable; [Formula: see text]; or [Formula: see text] for some prime power [Formula: see text].

scholarly journals EP-1661: Six degree set up errors of spine tumors assessed by image guided radiotherapy

2015 ◽  
Vol 115 ◽  
pp. S910
Author(s):  
P. Jiang ◽  
S. Zhou ◽  
J.J. Wang ◽  
R.J. Yang ◽  
Z.Y. Liu ◽  
...  
Keyword(s):  
Image Guided Radiotherapy ◽  
Image Guided ◽  
Spine Tumors ◽  
Degree Set ◽  
Set Up

scholarly journals On the discrepancy of random low degree set systems

2020 ◽  
Vol 57 (3) ◽  
pp. 695-705
Author(s):  
Nikhil Bansal ◽  
Raghu Meka
Keyword(s):  
Set Systems ◽  
Low Degree ◽  
Degree Set

scholarly journals Biregular Cages of Girth Five

10.37236/2594 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Marién Abreu ◽  
Gabriela Araujo-Pardo ◽  
Camino Balbuena ◽  
Domenico Labbate ◽  
Gloria López-Chávez
Keyword(s):  
Prime Power ◽  
Discrete Math ◽  
Regular Graphs ◽  
Minimum Order ◽  
Degree Set ◽  
Positive Integers

Let $2 \le r < m$ and $g$ be positive integers. An $(\{r,m\};g)$-graph (or biregular graph) is a graph with degree set $\{r,m\}$ and girth $g$, and an $(\{r,m\};g)$-cage (or biregular cage) is an $(\{r,m\};g)$-graph of minimum order $n(\{r,m\};g)$. If $m=r+1$, an $(\{r,m\};g)$-cage is said to be a semiregular cage.In this paper we generalize the reduction and graph amalgam operations from [M. Abreu,  G. Araujo-Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth $5$. Discrete Math. 312 (2012) 2832--2842] on the incidence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new families are $(\{r,2r-3\};5)$-cages for all $r=q+1$ with $q$ a prime power, and $(\{r,2r-5\};5)$-cages for all $r=q+1$ with $q$ a prime. The new semiregular cages are constructed for $r=5$ and $6$ with $31$ and $43$ vertices respectively.

scholarly journals Tripartite graphs with given degree set

2015 ◽  
Vol 7 (1) ◽  
pp. 72-106
Author(s):  
Antal Iványi ◽  
Shariefuddin Pirzada ◽  
Farooq A. Dar
Keyword(s):  
The Family ◽  
Degree Set ◽  
Tripartite Graphs

Abstract If k ≥ 1, then the global degree set of a k-partite graph G = (V1, V2, . . . , Vk, E) is the set of the distinct degrees of the vertices of G, while if k ≥ 2, then the distributed degree set of G is the family of the k degree sets of the vertices of the parts of G. We propose algorithms to construct bipartite and tripartite graphs with prescribed global and distributed degree sets consisting from arbitrary nonnegative integers. We also present a review of the similar known results on digraphs.

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