Odd neighborhood transversals on grid graphs

Cowen, Robert; Hechler, Stephen H.; Kennedy, J. Ward; Steinberg, Arthur

doi:10.1016/j.disc.2006.11.006

Cited by 21 publications

(25 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore P G (y) = ρ and Proposition 10 implies that y is a nonnegative eigenvector to ρ, which by Theorem 9 must be positive. In addition, equality holds in (5) for every k ∈ [n]; thus, − sign(x r k ) = sign(x k x i 2 · · · x i r ) (6) whenever a k,i 2 ,...,i r = 0. Since A is symmetric, we get…”

Section: Odd Transversals and H-eigenvaluesmentioning

confidence: 99%

See 1 more Smart Citation

Hypergraphs and hypermatrices with symmetric spectrum

Nikiforov

2017

Linear Algebra and its Applications

View full text Add to dashboard Cite

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric r-matrix A has a symmetric spectrum if and only if r is even and A is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs with symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu.Separately, similar results are obtained for the H-spectram of hypermatrices.

show abstract

Section: Odd Transversals and H-eigenvaluesmentioning

confidence: 99%

“…whenever a i 1 ,...,i r = 0, and thus r is even. Therefore, (6) implies that x i 1 · · · x i r < 0 whenever a i 1 ,...,i r = 0, and so the set of indices of the negative entries of (x 1 , . .…”

Section: Odd Transversals and H-eigenvaluesmentioning

confidence: 99%

Hypergraphs and hypermatrices with symmetric spectrum

Nikiforov

2017

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…In [13] it was proved that the tensor product of any number of simple graphs has a total perfect code if and only if each factor has a total perfect code. In [5,20] the grid graphs that have total perfect codes were characterized. In [24], lexicographic, strong, and disjunctive products of graphs admitting total perfect codes were characterized, and a similar result was also obtained for the cartesian product of any graph with the complete graph of two vertices.…”

Section: Introductionmentioning

confidence: 99%

Total perfect codes in Cayley graphs

Zhou

2016

Des. Codes Cryptogr.

View full text Add to dashboard Cite

A total perfect code in a graph $\Gamma$ is a subset $C$ of $V(\Gamma)$ such that every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. We give necessary and sufficient conditions for a conjugation-closed subset of a group to be a total perfect code in a Cayley graph of the group. As an application we show that a Cayley graph on an elementary abelian $2$-group admits a total perfect code if and only if its degree is a power of $2$. We also obtain necessary conditions for a Cayley graph of a group with connection set closed under conjugation to admit a total perfect code.Comment: This is the final version published in Designs, Codes and Cryptograph

show abstract

“…Total perfect codes were also discussed in a paper Gavlas and Schultz [25] under the name open efficient dominating sets and in a paper by Cowen et al [20], who called them exact transversals. The latter paper poses the problem of determining which grid graphs have total perfect codes and solves it in the case both dimensions of the grid graph are odd.…”

Section: Total Perfect Codesmentioning

confidence: 98%