scholarly journals Edge Even Graceful Labeling of Polar Grid Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 38 ◽  
Author(s):  
Salama Nagy Daoud
Keyword(s):  
Sufficient Conditions ◽  
Simple Graph ◽  
Necessary And Sufficient Conditions ◽  
Grid Graph ◽  
Graceful Labeling ◽  
Grid Graphs ◽  
Polar Grid ◽  
Necessary And Sufficient

Edge Even Graceful Labelingwas first defined byElsonbaty and Daoud in 2017. An edge even graceful labeling of a simple graph G with p vertices and q edges is a bijection f from the edges of the graph to the set { 2 , 4 , … , 2 q } such that, when each vertex is assigned the sum of all edges incident to it mod 2 r where r = max { p , q } , the resulting vertex labels are distinct. In this paper we proved necessary and sufficient conditions for the polar grid graph to be edge even graceful graph.

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 Hamiltonian Paths in Some Classes of Grid Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Fatemeh Keshavarz-Kohjerdi ◽  
Alireza Bagheri
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Paths ◽  
Grid Graphs ◽  
Hamiltonian Path Problem ◽  
Np Complete ◽  
Necessary And Sufficient ◽  
Linear Time Algorithms

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs.

scholarly journals Multigraph decomposition into multigraphs with two underlying edges

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Miri Priesler ◽  
Michael Tarsi
Keyword(s):  
Decision Problem ◽  
Sufficient Conditions ◽  
Simple Graph ◽  
Necessary And Sufficient Conditions ◽  
International Audience ◽  
Necessary And Sufficient ◽  
The Given

International audience Due to some intractability considerations, reasonable formulation of necessary and sufficient conditions for decomposability of a general multigraph G into a fixed connected multigraph H, is probably not feasible if the underlying simple graph of H has three or more edges. We study the case where H consists of two underlying edges. We present necessary and sufficient conditions for H-decomposability of G, which hold when certain size parameters of G lies within some bounds which depends on the multiplicities of the two edges of H. We also show this result to be "tight" in the sense that even a slight deviation of these size parameters from the given bounds results intractability of the corresponding decision problem.

scholarly journals The cordiality of the sum and union of two fourth power of paths and cycles

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Ashraf Elrokh ◽  
Aya Rabie
Keyword(s):  
Sufficient Conditions ◽  
Simple Graph ◽  
Necessary And Sufficient Conditions ◽  
Fourth Power ◽  
Necessary And Sufficient

AbstractA simple graph is called cordial if it has 0-1 labeling that satisfies certain conditions. In this paper, we examine the necessary and sufficient conditions for cordial labeling of the sum and union of two fourth power of paths and cycles.

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> 0 for eachj> 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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