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1
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C7-Decompositions of the Tensor Product of Complete Graphs

100%
Manikandan R. S., Paulraja P.
Discussiones Mathematicae Graph Theory
|
2017
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tom 37
|
nr 3
523-535
EN
In this paper we consider a decomposition of Km × Kn, where × denotes the tensor product of graphs, into cycles of length seven. We prove that for m, n ≥ 3, cycles of length seven decompose the graph Km × Kn if and only if (1) either m or n is odd and (2) 14 | m(m − 1)n(n − 1). The results of this paper together with the results of [Cp-Decompositions of some regular graphs, Discrete Math. 306 (2006) 429–451] and [C5-Decompositions of the tensor product of complete graphs, Australasian J. Combinatorics 37 (2007) 285–293], give necessary and sufficient conditions for the existence of a p-cycle decomposition, where p ≥ 5 is a prime number, of the graph Km × Kn.
2
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Wiener index of the tensor product of a path and a cycle

100%
Pattabiraman K., Paulraja P.
Discussiones Mathematicae Graph Theory
|
2011
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tom 31
|
nr 4
737-751
EN
The Wiener index, denoted by W(G), of a connected graph G is the sum of all pairwise distances of vertices of the graph, that is, $W(G) = ½Σ_{u,v ∈ V(G)} d(u,v)$. In this paper, we obtain the Wiener index of the tensor product of a path and a cycle.
3
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Harary Index of Product Graphs

80%
Pattabiraman K., Paulraja P.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
|
nr 1
17-33
EN
The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, the exact formulae for the Harary indices of tensor product G × Km0,m1,...,mr−1 and the strong product G⊠Km0,m1,...,mr−1 , where Km0,m1,...,mr−1 is the complete multipartite graph with partite sets of sizes m0,m1, . . . ,mr−1 are obtained. Also upper bounds for the Harary indices of tensor and strong products of graphs are estabilished. Finally, the exact formula for the Harary index of the wreath product G ○ G′ is obtained.
4
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Isomorphic components of direct products of bipartite graphs

80%
Hammack R.
Discussiones Mathematicae Graph Theory
|
2006
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tom 26
|
nr 2
231-248
EN
A standard result states the direct product of two connected bipartite graphs has exactly two components. Jha, Klavžar and Zmazek proved that if one of the factors admits an automorphism that interchanges partite sets, then the components are isomorphic. They conjectured the converse to be true. We prove the converse holds if the factors are square-free. Further, we present a matrix-theoretic conjecture that, if proved, would prove the general case of the converse; if refuted, it would produce a counterexample.
5
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Flat semilattices

70%
Grätzer G., Wehrung F.
Colloquium Mathematicum
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1999
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tom 79
|
nr 2
185-191
6
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On decomposition of k-tridiagonal ℓ-Toeplitz matrices and its applications

70%
Ohashi A., Sogabe T., Usuda T. S.
Special Matrices
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2015
|
tom 3
|
nr 1
EN
We consider a k-tridiagonal ℓ-Toeplitz matrix as one of generalizations of a tridiagonal Toeplitz matrix. In the present paper, we provide a decomposition of the matrix under a certain condition. By the decomposition, the matrix is easily analyzed since one only needs to analyze the small matrix obtained from the decomposition. Using the decomposition, eigenpairs and arbitrary integer powers of the matrix are easily shown as applications.
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