scholarly journals Unicyclic Graphs Whose Completely Regular Endomorphisms form a Monoid

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 240
Author(s):  
Rui Gu ◽  
Hailong Hou
Keyword(s):  
Sufficient Conditions ◽  
Unicyclic Graph ◽  
Necessary And Sufficient Conditions ◽  
Unicyclic Graphs ◽  
Completely Regular ◽  
Odd Cycle ◽  
Necessary And Sufficient

In this paper, completely regular endomorphisms of unicyclic graphs are explored. Let G be a unicyclic graph and let c E n d ( G ) be the set of all completely regular endomorphisms of G. The necessary and sufficient conditions under which c E n d ( G ) forms a monoid are given. It is shown that c E n d ( G ) forms a submonoid of E n d ( G ) if and only if G is an odd cycle or G = G ( n , m ) for some odd n ≥ 3 and integer m ≥ 1 .

scholarly journals Hamiltonian Cycles in Squares of Vertex-Unicyclic Graphs

1976 ◽  
Vol 19 (2) ◽  
pp. 169-172 ◽  
Author(s):  
Herbert Fleischner ◽  
Arthur M. Hobbs
Keyword(s):  
Sufficient Conditions ◽  
Unicyclic Graph ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Cycles ◽  
Unicyclic Graphs ◽  
Necessary And Sufficient

In this paper we determine necessary and sufficient conditions for the square of a vertex-unicyclic graph to be Hamiltonian. The conditions are simple and easily checked. Further, we show that the square of a vertex-unicyclic graph is Hamiltonian if and only if it is vertex-pancyclic.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

Some semigroup classes and congruences on Bruck-Reilly and generalized Bruck-Reilly ∗-extensions of monoids

2015 ◽  
Vol 08 (04) ◽  
pp. 1550075
Author(s):  
Seda Oğuz ◽  
Eylem G. Karpuz
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Completely Regular ◽  
Necessary And Sufficient

In this paper, we determine necessary and sufficient conditions for Bruck–Reilly and generalized Bruck–Reilly ∗-extensions of monoids to be unit regular, completely regular and orthodox. In addition, we characterize the congruences on generalized Bruck–Reilly ∗-extension of a monoid.

scholarly journals On maps between Stone-Cech compactifications induced by lattice homomorphisms

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2465-2474 ◽  
Author(s):  
Themba Dube
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lattice Homomorphism ◽  
Completely Regular ◽  
Zero Sets ◽  
Continuous Map ◽  
Necessary And Sufficient ◽  
Lattice Homomorphisms ◽  
Tychonoff Spaces

Broverman has shown that if X and Y are Tychonoff spaces and t:Z(Y)?Z(X) is a lattice homomorphism between the lattices of their zero-sets, then there is a continuous map ?: ?X ? ?Y induced by t. In this note we expound this idea and supplement Broverman?s results by first showing that this phenomenon holds in the category of completely regular frames. Among results we obtain, which were not considered by Broverman, are necessary and sufficient conditions (in terms of properties of the map t) for the induced map ? to be (i) the inclusion of a subspace, (ii) surjective, and (iii) irreducible. We show that if X and Y are pseudocompact then t pulls back z-ultrafilters to z-ultrafilters if and only if cl?X t(Z) = ?? [cl?YZ] for every Z ? Z(Y) if and only if t is ?-homomorphism.

scholarly journals Note on pointwise contractive projections

1993 ◽  
Vol 16 (4) ◽  
pp. 817-818
Author(s):  
L. M. Sanchez Ruiz ◽  
J. R. Ferrer Villanueva
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Compact Open Topology ◽  
Completely Regular ◽  
Contractive Projection ◽  
Necessary And Sufficient ◽  
Contractive Projections

LetC(X)be the space of real-valued continuous functions on a Hausdorff completely regular topological spaceX. endowed with the compact-open topology. In this paper necessary and sufficient conditions are given for a subspace ofC(X)to be the range of a pointwise contractive projection inC(X).

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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