Vector representations of graphs
TLDR
The least dimension d necessary for vector representations of graphs is studied as a function of G and of various restrictions placed upon the coordinates of the vectors and the values of the inner products.About:
This article is published in Discrete Mathematics.The article was published on 1989-11-01 and is currently open access. It has received 39 citations till now. The article focuses on the topics: Neighbourhood (graph theory) & Vertex (geometry).read more
Citations
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Linearly independent vertices and minimum semidefinite rank
Philip Hackney,Benjamin Harris,Margaret Lay,Lon H. Mitchell,Sivaram K. Narayan,Amanda Pascoe +5 more
TL;DR: In this article, the authors studied the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs, and provided a sufficient condition for when the vectors corresponding to a set of vertices of a given graph must be linearly independent in any vector representation of that graph, and conjecture that the resulting graph invariant is equal to minimum semi-definite rank.
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A combinatorial approach to complexity
Pavel Pudlák,Vojtech Rödl +1 more
TL;DR: If these graphs were explicitly constructed, they would have an explicit construction of Boolean functions of large complexity.
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Lower Bounds in Minimum Rank Problems
TL;DR: This work explores connections between OS-sets and a lower bound for minimum rank related to zero forcing sets as well as exhibit graphs for which the difference between the minimum semidefinite rank and these lower bounds can be arbitrarily large.
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Orthogonal Vector Coloring
TL;DR: Two vector analogues of list coloring along with their chromatic numbers are introduced and characterized and all graphs that have (vector) chromatic number two in each case are characterized.
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All generalized petersen graphs are unit-distance graphs
TL;DR: Bouwer et al. as discussed by the authors showed that every generalized Petersengraph admits a unit-distance representation in the Euclidean plane with a n-fold rotational symmetry, with the exception of the families I(n;j;j) and I(12m,m;m;5m).
References
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Book
Graph theory with applications
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
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Graph theory with applications (revised edition), by J. A. Bondy and U.S.R. Murty. Pp x, 264. £5·95 paperback. 1977. SBN 0 333 22694 1 (Macmillan)
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On the Shannon capacity of a graph
TL;DR: It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.
Book
Connectivity in Graphs
TL;DR: In this paper, a simple path is defined as a path that does not repeat any edges and is non-simple in the sense that it can be non-repeatable from x to y.