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- Matthew Booth, Philip Hackney, +8 authors Wendy Wang
- SIAM J. Matrix Analysis Applications
- 2008

- Gerald Haynes, Catherine Park, Amanda Schaeffer, Jordan Webster, Lon H. Mitchell
- Electr. J. Comb.
- 2010

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)

The Colin deVerdì ere parameters, µ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for µ the calculation depends on split subgraphs. For the case of µ… (More)

- Lon H. Mitchell
- Discrete Mathematics
- 2004

- Lon H. Mitchell, Sivaram K. Narayan, Ian Rogers, LON H. MITCHELL, SIVARAM K. NARAYAN
- 2017

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 Fd . 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)

- Francesco Barioli, Shaun M. Fallat, +4 authors SIVARAM K. NARAYAN
- 2017

Let G = (V, E) be a multigraph with no loops on the vertex set V = {1, 2, . . . , n}. Define S+(G) as the set of symmetric positive semidefinite matrices A = [aij ] with aij 6= 0, i 6= j, if ij ∈ E(G) is a single edge and aij = 0, i 6= j, if ij / ∈ E(G). Let M+(G) denote the maximum multiplicity of zero as an eigenvalue of A ∈ S+(G) and mr+(G) = |G|−M+(G)… (More)

- Jonathan Beagley, Lon H. Mitchell, +5 authors Andrew M. Zimmer
- 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)

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)

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)

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