On tensor product graphs

Sampathkumar, E.

doi:10.1017/s1446788700020619

Cited by 21 publications

(18 citation statements)

References 3 publications

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“…In the case of Markov chains on graphs described by adjacency matrices, the tensor product of transition matrices correspond to taking the graph tensor product [35] of the underlying graphs. One important class of graphs formed from a tensor product of graphs is the bipartite double cover of a given graph G. If the adjacency matrix of G is given by A, then the bipartite double cover of G is given by the adjacency matrix A ⊗ K 2 , where K 2 = 0 1 1 0 [36].…”

Section: Tensor Product Of Markov Chainsmentioning

confidence: 99%

Efficient quantum circuits for Szegedy quantum walks

Loke

Wang

2017

Annals of Physics

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A major advantage in using Szegedy's formalism over discrete-time and continuous-time quantum walks lies in its ability to define a unitary quantum walk on directed and weighted graphs. In this paper, we present a general scheme to construct efficient quantum circuits for Szegedy quantum walks that correspond to classical Markov chains possessing transformational symmetry in the columns of the transition matrix. In particular, the transformational symmetry criteria do not necessarily depend on the sparsity of the transition matrix, so this scheme can be applied to non-sparse Markov chains. Two classes of Markov chains that are amenable to this construction are cyclic permutations and complete bipartite graphs, for which we provide explicit efficient quantum circuit implementations. We also prove that our scheme can be applied to Markov chains formed by a tensor product. We also briefly discuss the implementation of Markov chains based on weighted interdependent networks. In addition, we apply this scheme to construct efficient quantum circuits simulating the Szegedy walks used in the quantum Pagerank algorithm for some classes of non-trivial graphs, providing a necessary tool for experimental demonstration of the quantum Pagerank algorithm.

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Section: Tensor Product Of Markov Chainsmentioning

confidence: 99%

Efficient quantum circuits for Szegedy quantum walks

Loke

Wang

2017

Annals of Physics

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“…It is possible to show that performing a random walk on the tensor product graph G × is equivalent to performing a simultaneous random walk on G 1 and G 2 . There are many other important properties derived from the generalized product between graphs [7], [22], [23]. Therefore, the tensor product is a very interesting candidate in the inexact graph matching context, especially in the graph kernels family [2], [19], [24].…”

Section: A Tensor Product Of Graphsmentioning

confidence: 99%

Parallel algorithms for tensor product-based inexact graph matching

Livi

Rizzi

2012

The 2012 International Joint Conference on Neural Networks (IJCNN)

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In this paper we face the inexact graph matching problem from the parallel algorithms viewpoint. After a brief introduction of both graph matching and parallel computing contexts, we discuss a specific method of performing inexact graph matching based on the well known tensor product operator. We analyze the problem using two parallel computing models, following different algorithmic strategies, and performing also an experimental evaluation. The aim of this paper is to provide modeling and algorithmic strategies to extend inexact graph matching methods to graphs of high order and size, conceiving the computational problem in the more wider context of graphbased Pattern Recognition and Soft Computing systems. As a whole, the obtained results encourage more effort on this direction.

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“…The symmetric digraph D ( A ( G ) ) is obtained from G by replacing each edge of G by a symmetric pairzf arcs (a loop by a directed loop) and is denoted by 6. The bigraph B(G) is readily seen to be the graph K, x G, the Kronecker product [24] (also called tensor product [20] and conjunction [13]) of the graphs K, and G. Given a digraph D the underlying graph G ( D ) of D is the graph (possibly a multigraph) obtained by disregarding the orientation of the arcs in D. Thus each arc (vi, u j ) of D becomes an edge vivj of G ( D ) . In Figure 3 we show Lhe graph C, [the (undirectei) cycle, of length 41, the syrnmuetric digraph C,, the underlying graph G(C,) of C,, and the bigraph B(C,).…”

Section: N) G ( a )mentioning

confidence: 99%

“…permutation matrices P and Q such that PAQ is a symmetric matrix. Equivalently, D ( B ) is symmetric iff B can be represented in the form K , x G for some graph G. These graphs have been characterized by Sampathkumar [20]. An alternative characterization is the following: Theorem 2.1.…”

Section: N) G ( a )mentioning

confidence: 99%

Bigraphs versus digraphs via matrices

Brualdi¹,

Harary

Miller

1980

Journal of Graph Theory

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It was observed by Dulmage and Mendelsohn in their work on matrix reducibility that there is a one-to-one correspondence between bigraphs and digraphs determined by the utilization of the adjacency matrix. In this semiexpository paper we explore the interaction between this correspondence and a theory of matrix decomposability that is developed in several different articles. These results include: (a) a characterization of those bipartite graphs that can be labeled so that the resulting digraph is symmetric; (b) a criterion for the bigraph of a symmetric digraph to be connected; (c) a necessary and sufficient condition for a square binary matrix to be fully indecomposable in terms of its associated bigraph, and (d) matrix criteria for a digraph to be strongly, unilaterally, or weakly connected. We close with an unsolved extermal problem on the number of components of the bigraph of various orientations of a given graph. This leads to new amusing characterizations of trees and bigraphs.Dedicated to the graph-theoretic partnership of Lloyd Dulmage and Nathan Mendelsohn.

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On tensor product graphs

Abstract: The tensor product G ⊕ H of graphs G and H is the graph with point set V(G) × V(H) where (υ1, ν1) adj (υ2, ν2) if, and only if, u1 adj υ2 and ν1 adj ν2. We obtain a characterization of graphs of the form G ⊕ H where G or H is K2.

Cited by 21 publications

References 3 publications

Efficient quantum circuits for Szegedy quantum walks

Efficient quantum circuits for Szegedy quantum walks

Parallel algorithms for tensor product-based inexact graph matching

Bigraphs versus digraphs via matrices

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