scholarly journals Restricted triangulation on circulant graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 358-369 ◽  
Author(s):  
Niran Abbas Ali ◽  
Adem Kilicman ◽  
Hazim Michman Trao
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Existence Problem ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Value Set ◽  
Vertex Set ◽  
Necessary And Sufficient

AbstractThe restricted triangulation existence problem on a given graph decides whether there exists a triangulation on the graph’s vertex set that is restricted with respect to its edge set. Let G = C(n, S) be a circulant graph on n vertices with jump value set S. We consider the restricted triangulation existence problem for G. We determine necessary and sufficient conditions on S for which G admitting a restricted triangulation. We characterize a set of jump values S(n) that has the smallest cardinality with C(n, S(n)) admits a restricted triangulation. We present the measure of non-triangulability of Kn − G for a given G.

scholarly journals More on the annihilator-ideal graph of a commutative ring

2019 ◽  
Vol 18 (08) ◽  
pp. 1950160
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Simple Graph ◽  
Necessary And Sufficient Conditions ◽  
Star Graph ◽  
Annihilator Ideal ◽  
Vertex Set ◽  
Necessary And Sufficient

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph [Formula: see text] (a well known graph with the same vertices and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]) associated with [Formula: see text]. All rings whose [Formula: see text] and [Formula: see text] are characterized. Among other results, we obtain necessary and sufficient conditions under which [Formula: see text] is a star graph.

scholarly journals Classification of Efficient Total Domination Sets of Circulant Graphs of Degree 5

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1944
Author(s):  
Young Soo Soo Kwon ◽  
Moo Young Young Sohn
Keyword(s):  
Dominating Set ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Total Domination ◽  
Circulant Graphs ◽  
Total Dominating Set ◽  
Necessary And Sufficient

An efficient total dominating set D of a graph G is a vertex subset such that every vertex of G has exactly one neighbor in the set D. In this paper, we give necessary and sufficient conditions for the existence of efficient total domination sets of circulant graphs whose degree is 5 and classify these sets.

scholarly journals On the Number of Quasi-Kernels in Digraphs

2001 ◽  
Vol 8 (7) ◽  
Author(s):  
Gregory Gutin ◽  
Khee Meng Koh ◽  
Eng Guan Tay ◽  
Anders Yeo
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Vertex Set ◽  
Necessary And Sufficient

<p>A vertex set X of a digraph D = (V,A) is a kernel if X is independent<br /> (i.e., all pairs of distinct vertices of X are non-adjacent) and for<br />every v in V − X there exists x in X such that vx in A. A vertex set<br />X of a digraph D = (V,A) is a quasi-kernel if X is independent and<br />for every v in V − X there exist w in V − X, x in X such that either<br />vx in A or vw,wx in A: In 1994, Chvatal and Lovasz proved that every<br />digraph has a quasi-kernel. In 1996, Jacob and Meyniel proved that, <br />if  a digraph D has no kernel, then D contains at least three quasi-kernels.</p><p>We characterize digraphs with exactly one and two quasi-kernels, and,<br />thus, provide necessary and sufficient conditions for a digraph to have<br />at least three quasi-kernels. In particular, we prove that every strong<br />digraph of order at least three, which is not a 4-cycle, has at least<br />three quasi-kernels. We conjecture that every digraph with no sink<br />has a pair of disjoint quasi-kernels and provide some support to this<br />conjecture.</p>

scholarly journals On the scores and degrees in hypertournaments

2015 ◽  
Vol 7 (2) ◽  
pp. 200-215
Author(s):  
Shariefuddin Pirzada ◽  
Rameez Raja ◽  
Antal Iványi
Keyword(s):  
Mathematical Analysis ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Arrangement ◽  
Degree Of A Vertex ◽  
Degree Structure ◽  
Vertex Set ◽  
Necessary And Sufficient

Abstract A k-hypertournament H = (V, A), where V is the vertex set and A is an arc set, is a complete k-hypergraph with each k-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a k-hypertournament, the score si(losing score ri) of a vertex is the number of edges containing vi in which vi is not the last element(in which vi is the last element) and the total score of a vertex vi is ti = si − ri. For v ∈ V we denote $d_H^ + = \sum\limits_{a \in H} {\rho (v,a)} $ (or simply d+(v)) the degree of a vertex where, ρ(v, a) is k − i if v ∈ a ∈ A and v is the ith entry in a, otherwise zero. In this paper, we obtain necessary and sufficient conditions for a k-hypertournament to be degree regular. We use the inequalities of Holder and Chebyshev from mathematical analysis to study the score and degree structure of the k-hypertournaments.

Square element graph of square-subtract rings

2021 ◽  
pp. 2150142
Author(s):  
Bijon Biswas ◽  
S. Kar ◽  
M. K. Sen
Keyword(s):  
Special Class ◽  
Undirected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Vertex Set ◽  
Necessary And Sufficient

If [Formula: see text] is a ring then the square element graph [Formula: see text] is the simple undirected graph whose vertex set consists of all non-zero elements of [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text] for some [Formula: see text]. In this paper, we provide some necessary and sufficient conditions for the connectedness of [Formula: see text], where [Formula: see text] is a ring with identity. We mainly characterize some special class of ring [Formula: see text] which we call square-subtract ring for which the graph [Formula: see text] is connected.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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