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Homogeneous Cartesian products [PDF]
Australas. J Comb., 2002 Summary: A graph \(G\) is 1-homogeneous if certain isomorphisms between similarly embedded induced subgraphs of \(G\) extend to automorphisms of \(G\). We show that the only connected composite 1-homogeneous graphs are the cube, and \(K_n\times K_2\) and \(K_n\times K_n\) with \(n\geq 2\).
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Panconnectivity of Cartesian product graphs
The Journal of Supercomputing, 2009A graph G of order n (?2) is said to be panconnected if for each pair (x,y) of vertices of G there exists an xy-path of length ? for each ? such that d G (x,y)???n?1, where d G (x,y) denotes the length of a shortest xy-path in G. In this paper, we consider the panconnectivity of Cartesian product graphs.
You Lu 0002, Jun-Ming Xu 0001
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1970
When we say in analytic geometry that a point has co-ordinates (x, y), the order in which x and y occur, in the symbol (x, y), is important: (1, 2) ≠ (2, 1). For this reason we call (x, y) an ordered pair. Moreover, x and y come from sets; in this case x, y ∈ R. This idea can be generalizedf as follows. Let 𝒰 be a universe.
H. B. Griffiths, P. J. Hilton
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When we say in analytic geometry that a point has co-ordinates (x, y), the order in which x and y occur, in the symbol (x, y), is important: (1, 2) ≠ (2, 1). For this reason we call (x, y) an ordered pair. Moreover, x and y come from sets; in this case x, y ∈ R. This idea can be generalizedf as follows. Let 𝒰 be a universe.
H. B. Griffiths, P. J. Hilton
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Radicals commuting with cartesian products
Archiv der Mathematik, 1998Given an Abelian group \(G\), the group radical is defined by \(R_G(X)=\bigcap\{\text{Ker }\phi\mid\phi\colon X\to G\}\), for Abelian groups \(X\). This radical does not always commute with infinite direct products (for instance, when \(G=\mathbb{Q}\), it turns into the torsion radical).
Corner, A. L. S., Göbel, Rüdiger
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The determining number of a Cartesian product
Journal of Graph Theory, 2009AbstractA set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that if G=G□⋅□G is the prime factor decomposition of a connected graph then Det(G)=max{Det(G)}.
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Controllability of Cartesian Product Signed Networks
IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2023Junjie Huang, Housheng Su
exaly