Decompositions of some regular graphs into unicyclic graphs of order five

2019 ◽  
Vol 11 (04) ◽  
pp. 1950042 ◽  
Author(s):  
P. Paulraja ◽  
T. Sivakaran
Keyword(s):  
Necessary Conditions ◽  
Discrete Math ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Unicyclic Graphs ◽  
Multipartite Graphs ◽  
Complete Multipartite ◽  
Necessary And Sufficient ◽  
Product Of Graphs

For a graph [Formula: see text] and a subgraph [Formula: see text] of [Formula: see text] an [Formula: see text]-decomposition of [Formula: see text] is a partition of the edge set of [Formula: see text] into subsets [Formula: see text] [Formula: see text] such that each [Formula: see text] induces a graph isomorphic to [Formula: see text] It is proved that the necessary conditions are sufficient for the existence of an [Formula: see text]-decomposition of the graph [Formula: see text] where [Formula: see text] is any simple connected unicyclic graph of order five, × denotes the tensor product of graphs and [Formula: see text] denotes the multiplicity of the edges. In fact, using the above characterization, a necessary and sufficient condition for the graph [Formula: see text] [Formula: see text] and [Formula: see text] to admit an [Formula: see text]-decomposition is obtained. Similar results for the complete graphs and complete multipartite graphs are proved in: [J.-C. Bermond et al. [Formula: see text]-decomposition of [Formula: see text], where [Formula: see text] has four vertices or less, Discrete Math. 19 (1977) 113–120, J.-C. Bermond et al. Decomposition of complete graphs into isomorphic subgraphs with five verices, Ars Combin. 10 (1980) 211–254, M. H. Huang, Decomposing complete equipartite graphs into connected unicyclic graphs of size five, Util. Math. 97 (2015) 109–117].

scholarly journals Families of Integral Cographs within a Triangular Array

2020 ◽  
Vol 8 (1) ◽  
pp. 257-273
Author(s):  
Hsin-Yun Ching ◽  
Rigoberto Flórez ◽  
Antara Mukherjee
Keyword(s):  
Triangular Array ◽  
Sufficient Condition ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Empty Graph ◽  
Complete Product ◽  
Necessary And Sufficient

AbstractThe determinant Hosoya triangle, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 2 gives rise to three infinite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sufficient condition for a graph from these families to be integral.Some features of these graphs are: they are integral cographs, all graphs have at most five distinct eigenvalues, all graphs are either d-regular graphs with d =2, 4, 6, . . . or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.

scholarly journals \(C_{6}\)-decompositions of the tensor product of complete graphs

2020 ◽  
Vol 3 (3) ◽  
pp. 62-65
Author(s):  
Abolape Deborah Akwu ◽  
◽  
Opeyemi Oyewumi ◽  
Keyword(s):  
Tensor Product ◽  
Disjoint Union ◽  
Necessary Conditions ◽  
Finite Graph ◽  
Complete Graphs ◽  
Complete Multipartite Graph ◽  
Edge Disjoint ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Product Of Graphs

Let \(G\) be a simple and finite graph. A graph is said to be decomposed into subgraphs \(H_1\) and \(H_2\) which is denoted by \(G= H_1 \oplus H_2\), if \(G\) is the edge disjoint union of \(H_1\) and \(H_2\). If \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\), where \(H_1\), \(H_2\), ..., \(H_k\) are all isomorphic to \(H\), then \(G\) is said to be \(H\)-decomposable. Furthermore, if \(H\) is a cycle of length \(m\) then we say that \(G\) is \(C_m\)-decomposable and this can be written as \(C_m|G\). Where \( G\times H\) denotes the tensor product of graphs \(G\) and \(H\), in this paper, we prove that the necessary conditions for the existence of \(C_6\)-decomposition of \(K_m \times K_n\) are sufficient. Using these conditions it can be shown that every even regular complete multipartite graph \(G\) is \(C_6\)-decomposable if the number of edges of \(G\) is divisible by \(6\).

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

scholarly journals Syarat Perlu dan Cukup untuk Keterbatasan Potensial Riesz di Ruang Morrey Klasik

2013 ◽  
Vol 14 (3) ◽  
pp. 227
Author(s):  
Mohammad Imam Utoyo ◽  
Basuki Widodo ◽  
Toto Nusantara ◽  
Suhariningsih Suhariningsih
Keyword(s):  
Characteristic Function ◽  
Morrey Space ◽  
Necessary Conditions ◽  
Riesz Potential ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Research Results ◽  
Necessary And Sufficient

This script was aimed to determine the necessary conditions for boundedness of Riesz potential in the classical Morrey space. If these results are combined with previous research results will be obtained the necessary and sufficient condition for boundedness of Riesz potential. This necessary condition is obtained through the use of characteristic function as one member of the classical Morrey space.

Clairaut Riemannian maps whose total manifolds admit a Ricci soliton

2021 ◽  
Author(s):  
Akhilesh Yadav ◽  
Kiran Meena
Keyword(s):  
Vector Field ◽  
Necessary Conditions ◽  
Curvature Vector ◽  
Ricci Soliton ◽  
Necessary Condition ◽  
Geodesic Curve ◽  
Necessary And Sufficient Condition ◽  
Gradient Ricci Soliton ◽  
Riemannian Map ◽  
Necessary And Sufficient

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field [Formula: see text] to be conformal, where [Formula: see text] is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of [Formula: see text] then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.

Necessary and sufficient conditions for the two-point phase probability function of two-phase random media

2008 ◽  
Vol 464 (2095) ◽  
pp. 1761-1779 ◽  
Author(s):  
John A Quintanilla
Keyword(s):  
Random Media ◽  
Probability Function ◽  
Recent Literature ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Condition ◽  
Two Phase ◽  
Decomposition Formula ◽  
Positive Matrices ◽  
Necessary And Sufficient

Constructing realizations of random media with a specified two-point phase probability function S 2 has attracted considerable attention in the recent literature. However, little is known about conditions under which a prescribed S 2 is realizable. The only known necessary and sufficient condition, due to McMillan, involves a class of square matrices, called corner-positive matrices, about which almost nothing is known except their definition. As a result, McMillan's theorem has gone mostly unused in the literature for over 50 years. In this paper, we present a general decomposition formula for corner-positive matrices, which allows McMillan's theorem to be written in a significantly more tractable and testable form. We also connect McMillan's theorem with many known but heretofore unrelated necessary conditions on S 2 , extending many of these conditions.

scholarly journals Even Cycles and Even 2-Factors in the Line Graph of a Simple Graph

10.37236/5660 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Arrigo Bonisoli ◽  
Simona Bonvicini
Keyword(s):  
Perfect Matching ◽  
Line Graph ◽  
Simple Graph ◽  
Connected Components ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Cubic Graph ◽  
Cycle Decomposition ◽  
Odd Degree ◽  
Necessary And Sufficient

Let $G$ be a connected graph with an even number of edges. We show that if the subgraph of $G$ induced by the vertices of odd degree has a perfect matching, then the line graph of $G$ has a $2$-factor whose connected components are cycles of even length (an even $2$-factor). For a cubic graph $G$, we also give a necessary and sufficient condition so that the corresponding line graph $L(G)$ has an even cycle decomposition of index $3$, i.e., the edge-set of $L(G)$ can be partitioned into three $2$-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index $m$ in $2d$-regular graphs is also addressed.

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