scholarly journals On (2-d)-Kernels in the Tensor Product of Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 230
Author(s):  
Paweł Bednarz
Keyword(s):  
Tensor Product ◽  
Symmetric Distribution ◽  
Complete Graphs ◽  
Product Of Graphs

In this paper, we study the existence, construction and number of (2-d)-kernels in the tensor product of paths, cycles and complete graphs. The symmetric distribution of (2-d)-kernels in these products helps us to characterize them. Among others, we show that the existence of (2-d)-kernels in the tensor product does not require the existence of a (2-d)-kernel in their factors. Moreover, we determine the number of (2-d)-kernels in the tensor product of certain factors using Padovan and Perrin numbers.

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 \(6\).

Some classifications of graphs with respect to a set adjacency relation

2020 ◽  
pp. 2050089
Author(s):  
G. Chiaselotti ◽  
T. Gentile ◽  
F. G. Infusino
Keyword(s):  
Computational Complexity ◽  
Tensor Product ◽  
Undirected Graph ◽  
Complete Graphs ◽  
Adjacency Relation ◽  
Vertex Set ◽  
Complete Bipartite Graphs ◽  
Graph Families ◽  
Graded Lattice ◽  
Product Of Graphs

For any finite simple undirected graph [Formula: see text], we consider the binary relation [Formula: see text] on the powerset [Formula: see text] of its vertex set given by [Formula: see text] if [Formula: see text], where [Formula: see text] denotes the neighborhood of a vertex [Formula: see text]. We call the above relation set adiacence dependency (sa)-dependency of [Formula: see text]. With the relation [Formula: see text] we associate an intersection-closed family [Formula: see text] of vertex subsets and the corresponding induced lattice [Formula: see text], which we call sa-lattice of [Formula: see text]. Through the equality of sa-lattices, we introduce an equivalence relation [Formula: see text] between graphs and propose three different classifications of graphs based on such a relation. Furthermore, we determine the sa-lattice for various graph families, such as complete graphs, complete bipartite graphs, cycles and paths and, next, we study such a lattice in relation to the Cartesian and the tensor product of graphs, verifying that in most cases it is a graded lattice. Finally, we provide two algorithms, namely, the T-DI ALGORITHM and the O-F ALGORITHM, in order to provide two different computational ways to construct the sa-lattice of a graph. For the O-F ALGORITHM we also determine its computational complexity.

scholarly journals On Hamilton cycle decompositions of the tensor product of complete graphs

2003 ◽  
Vol 268 (1-3) ◽  
pp. 49-58 ◽  
Author(s):  
R. Balakrishnan ◽  
J.-C. Bermond ◽  
P. Paulraja ◽  
M.-L. Yu
Keyword(s):  
Tensor Product ◽  
Hamilton Cycle ◽  
Complete Graphs ◽  
Cycle Decompositions

scholarly journals Hamilton cycles in tensor product of graphs

1998 ◽  
Vol 186 (1-3) ◽  
pp. 1-13 ◽  
Author(s):  
R. Balakrishnan ◽  
P. Paulraja
Keyword(s):  
Tensor Product ◽  
Hamilton Cycles ◽  
Product Of Graphs

SIMPLE GRAPHOIDAL COVER ON TENSOR PRODUCT OF GRAPHS

2019 ◽  
Vol 22 (1) ◽  
pp. 1-40
Author(s):  
G. Venkat Narayanan ◽  
J. Suresh Suseela ◽  
R. Kala
Keyword(s):  
Tensor Product ◽  
Product Of Graphs

Downhill domination in the tensor product of graphs

2019 ◽  
Vol 13 (12) ◽  
pp. 555-564
Author(s):  
Hilbert R. Acosta ◽  
Rolito G. Eballe ◽  
Isagani S. Cabahug Jr
Keyword(s):  
Tensor Product ◽  
Product Of Graphs

scholarly journals Gregarious kite factorization of tensor product of complete graphs

2020 ◽  
Vol 40 (1) ◽  
pp. 7
Author(s):  
A. Tamil Elakkiya ◽  
Appu Muthusamy
Keyword(s):  
Tensor Product ◽  
Complete Graphs
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