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A vector coloring of a graph is an assignment of a vector to each vertex where the presence or absence of an edge between two vertices dictates the value of the inner product of the corresponding vectors. In this paper, we obtain results on orthogonal vector coloring, where adjacent vertices must be assigned orthogonal vectors. We introduce two vector… (More)

- Jonathan Beagley, Lon H. Mitchell, Sivaram K. Narayanc, Eileen Radzwion, Sara P. Rimer, Rachael Tomasino +2 others
- 2013

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: The real (complex) minimum semidefinite rank of a graph is the minimum rank among all… (More)

The minimum vector rank (mvr) of a graph over a field F is the smallest d for which a faithful vector representation of G exists in F d. For simple graphs, minimum semidefinite rank (msr) and minimum vector rank differ by exactly the number of isolated vertices. We explore the relationship between msr and mvr for multigraphs and show that a result linking… (More)

Let L(G) be the Laplacian matrix of a simple graph G. The characteristic valuation associated with the algebraic connectivity a(G) is used in classifying trees as Type I and Type II. We show a tree T is Type I if and only if its algebraic connectivity a(T) belongs to the spectrum of some branch B of T. I am very greatful to Dr. Narayan for his devotion to… (More)

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